%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LCL379-1 : TPTP v9.3.1. Released v2.3.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 11:49:42 AM UTC 2026
% Result : Unsatisfiable 39.16s 5.45s
% Output : Proof 56.94s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : LCL379-1 : TPTP v9.3.1. Released v2.3.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.10/0.36 % Computer : n005.cluster.edu
% 0.10/0.36 % Model : x86_64 x86_64
% 0.10/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.36 % Memory : 8046.5625MB
% 0.10/0.36 % OS : Linux 6.8.0-71-generic
% 0.10/0.36 % CPULimit : 300
% 0.10/0.36 % WCLimit : 300
% 0.10/0.36 % DateTime : Sun Sep 27 15:38:01 UTC 2026
% 0.10/0.37 % CPUTime :
% 0.10/0.37 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 39.16/5.45 Command-line arguments: --no-flatten-goal
% 39.16/5.45
% 39.16/5.45 % SZS status Unsatisfiable
% 39.16/5.45
% 50.35/7.74 % SZS output start Proof
% 50.35/7.74 Axiom 1 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 50.35/7.74 Axiom 2 (cn_3): is_a_theorem(implies(X, implies(not(X), Y))) = true.
% 50.35/7.74 Axiom 3 (cn_2): is_a_theorem(implies(implies(not(X), X), X)) = true.
% 50.35/7.74 Axiom 4 (condensed_detachment): ifeq(is_a_theorem(implies(X, Y)), true, ifeq(is_a_theorem(X), true, is_a_theorem(Y), true), true) = true.
% 51.04/7.80 Axiom 5 (cn_1): is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
% 51.04/7.80
% 51.04/7.80 Goal 1 (prove_cn_38): is_a_theorem(implies(implies(x, not(x)), not(x))) = true.
% 51.04/7.80 Proof:
% 51.04/7.80 is_a_theorem(implies(implies(x, not(x)), not(x)))
% 51.04/7.80 = { by axiom 1 (ifeq_axiom) R->L }
% 51.04/7.80 ifeq(true, true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.80 = { by axiom 4 (condensed_detachment) R->L }
% 51.04/7.80 ifeq(ifeq(is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.80 = { by axiom 1 (ifeq_axiom) R->L }
% 51.04/7.80 ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.80 = { by axiom 4 (condensed_detachment) R->L }
% 51.04/7.81 ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), not(x)), implies(implies(not(not(x)), not(x)), not(x))), implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))))), true, ifeq(is_a_theorem(implies(implies(not(x), not(x)), implies(implies(not(not(x)), not(x)), not(x)))), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.81 = { by axiom 5 (cn_1) }
% 51.04/7.81 ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(x), not(x)), implies(implies(not(not(x)), not(x)), not(x)))), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.81 = { by axiom 1 (ifeq_axiom) }
% 51.04/7.81 ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(x), not(x)), implies(implies(not(not(x)), not(x)), not(x)))), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.81 = { by axiom 1 (ifeq_axiom) R->L }
% 51.04/7.81 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(not(not(x)), not(x)), not(x)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.81 = { by axiom 5 (cn_1) R->L }
% 51.04/7.81 ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(not(x), not(x)), implies(implies(not(not(x)), not(x)), not(x))))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(not(not(x)), not(x)), not(x)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.81 = { by axiom 1 (ifeq_axiom) R->L }
% 51.04/7.81 ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(not(x), not(x)), implies(implies(not(not(x)), not(x)), not(x))))), true, ifeq(true, true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(not(not(x)), not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.81 = { by axiom 3 (cn_2) R->L }
% 51.04/7.81 ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(not(x), not(x)), implies(implies(not(not(x)), not(x)), not(x))))), true, ifeq(is_a_theorem(implies(implies(not(not(x)), not(x)), not(x))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(not(not(x)), not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.81 = { by axiom 4 (condensed_detachment) }
% 51.04/7.81 ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.81 = { by axiom 1 (ifeq_axiom) }
% 51.04/7.81 ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.82 = { by axiom 1 (ifeq_axiom) R->L }
% 51.04/7.82 ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.82 = { by axiom 4 (condensed_detachment) R->L }
% 51.04/7.82 ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x)))), implies(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))))), true, ifeq(is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.82 = { by axiom 5 (cn_1) }
% 51.04/7.82 ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.82 = { by axiom 1 (ifeq_axiom) }
% 51.04/7.82 ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.82 = { by axiom 5 (cn_1) }
% 51.04/7.82 ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.82 = { by axiom 1 (ifeq_axiom) }
% 51.04/7.82 ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(x, not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(x, not(x)), implies(not(not(x)), not(x))), implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x))))), true), true), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.82 = { by axiom 4 (condensed_detachment) }
% 51.04/7.82 ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.82 = { by axiom 1 (ifeq_axiom) }
% 51.04/7.82 ifeq(ifeq(is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.82 = { by axiom 1 (ifeq_axiom) R->L }
% 51.04/7.82 ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.82 = { by axiom 4 (condensed_detachment) R->L }
% 51.04/7.82 ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)), implies(not(not(x)), implies(not(x), x)))), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.83 = { by axiom 1 (ifeq_axiom) R->L }
% 51.04/7.83 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)), implies(not(not(x)), implies(not(x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.83 = { by axiom 5 (cn_1) R->L }
% 51.04/7.83 ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(x))), not(not(x)))), implies(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)), implies(not(not(x)), implies(not(x), x))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)), implies(not(not(x)), implies(not(x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.83 = { by axiom 1 (ifeq_axiom) R->L }
% 51.04/7.83 ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(x))), not(not(x)))), implies(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)), implies(not(not(x)), implies(not(x), x))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)), implies(not(not(x)), implies(not(x), x)))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.83 = { by axiom 2 (cn_3) R->L }
% 51.04/7.83 ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(x))), not(not(x)))), implies(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)), implies(not(not(x)), implies(not(x), x))))), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(not(not(x))), not(not(x))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)), implies(not(not(x)), implies(not(x), x)))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.83 = { by axiom 4 (condensed_detachment) }
% 51.04/7.83 ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.83 = { by axiom 1 (ifeq_axiom) }
% 51.04/7.83 ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.83 = { by axiom 1 (ifeq_axiom) R->L }
% 51.04/7.83 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.83 = { by axiom 4 (condensed_detachment) R->L }
% 51.04/7.83 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.84 = { by axiom 1 (ifeq_axiom) R->L }
% 51.04/7.84 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.84 = { by axiom 4 (condensed_detachment) R->L }
% 51.04/7.84 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x)))))), implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x)))))))), true, ifeq(is_a_theorem(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.84 = { by axiom 1 (ifeq_axiom) R->L }
% 51.04/7.85 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x)))))), implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.85 = { by axiom 5 (cn_1) R->L }
% 51.04/7.85 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(x), implies(not(not(x)), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x)))))), implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x)))))), implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.85 = { by axiom 1 (ifeq_axiom) R->L }
% 51.04/7.85 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(x), implies(not(not(x)), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x)))))), implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x)))))), implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.85 = { by axiom 2 (cn_3) R->L }
% 51.04/7.86 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(x), implies(not(not(x)), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x)))))), implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))))), true, ifeq(is_a_theorem(implies(not(x), implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x)))))), implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.86 = { by axiom 4 (condensed_detachment) }
% 51.04/7.86 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.86 = { by axiom 1 (ifeq_axiom) }
% 51.04/7.86 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.86 = { by axiom 1 (ifeq_axiom) R->L }
% 51.04/7.86 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.87 = { by axiom 5 (cn_1) R->L }
% 51.04/7.87 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(not(x))), implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x)))))))), true, is_a_theorem(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.87 = { by axiom 1 (ifeq_axiom) R->L }
% 51.04/7.87 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(not(x))), implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x)))))))), true, ifeq(true, true, is_a_theorem(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.88 = { by axiom 3 (cn_2) R->L }
% 51.04/7.88 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(not(x))), implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x)))))))), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), not(not(x)))), true, is_a_theorem(implies(implies(not(not(x)), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.88 = { by axiom 4 (condensed_detachment) }
% 51.04/7.88 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.88 = { by axiom 1 (ifeq_axiom) }
% 51.04/7.88 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.88 = { by axiom 1 (ifeq_axiom) R->L }
% 51.04/7.88 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.88 = { by axiom 5 (cn_1) R->L }
% 51.04/7.88 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x)))))))), true, ifeq(is_a_theorem(implies(not(x), implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(x), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.88 = { by axiom 4 (condensed_detachment) }
% 51.04/7.89 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.89 = { by axiom 1 (ifeq_axiom) }
% 51.04/7.89 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.89 = { by axiom 1 (ifeq_axiom) R->L }
% 51.04/7.89 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.04/7.89 = { by axiom 4 (condensed_detachment) R->L }
% 51.04/7.90 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.75/7.90 = { by axiom 1 (ifeq_axiom) R->L }
% 51.75/7.90 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.75/7.90 = { by axiom 4 (condensed_detachment) R->L }
% 51.75/7.91 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))))), true, ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.75/7.91 = { by axiom 5 (cn_1) }
% 51.75/7.91 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.75/7.92 = { by axiom 1 (ifeq_axiom) }
% 51.75/7.92 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.75/7.93 = { by axiom 1 (ifeq_axiom) R->L }
% 51.75/7.93 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.75/7.93 = { by axiom 5 (cn_1) R->L }
% 51.75/7.94 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.75/7.94 = { by axiom 1 (ifeq_axiom) R->L }
% 51.75/7.94 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true, ifeq(true, true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.75/7.95 = { by axiom 3 (cn_2) R->L }
% 51.75/7.95 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.75/7.95 = { by axiom 4 (condensed_detachment) }
% 51.75/7.96 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.75/7.96 = { by axiom 1 (ifeq_axiom) }
% 51.75/7.97 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.75/7.97 = { by axiom 1 (ifeq_axiom) R->L }
% 51.75/7.98 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.75/7.98 = { by axiom 4 (condensed_detachment) R->L }
% 51.75/7.99 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), implies(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 51.75/7.99 = { by axiom 5 (cn_1) }
% 51.75/8.00 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 52.47/8.00 = { by axiom 1 (ifeq_axiom) }
% 52.47/8.00 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 52.47/8.01 = { by axiom 5 (cn_1) }
% 52.47/8.01 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 52.47/8.01 = { by axiom 1 (ifeq_axiom) }
% 52.47/8.01 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 52.47/8.02 = { by axiom 4 (condensed_detachment) }
% 52.47/8.02 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 52.47/8.02 = { by axiom 1 (ifeq_axiom) }
% 52.47/8.02 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 52.47/8.02 = { by axiom 1 (ifeq_axiom) R->L }
% 52.47/8.02 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 52.47/8.03 = { by axiom 4 (condensed_detachment) R->L }
% 52.47/8.03 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true, ifeq(is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 52.47/8.03 = { by axiom 5 (cn_1) }
% 52.47/8.03 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 52.47/8.04 = { by axiom 1 (ifeq_axiom) }
% 52.47/8.04 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 52.47/8.04 = { by axiom 1 (ifeq_axiom) R->L }
% 52.47/8.04 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 52.47/8.05 = { by axiom 4 (condensed_detachment) R->L }
% 52.47/8.05 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 52.47/8.06 = { by axiom 1 (ifeq_axiom) R->L }
% 52.47/8.06 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 52.47/8.06 = { by axiom 5 (cn_1) R->L }
% 52.47/8.07 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 52.47/8.07 = { by axiom 1 (ifeq_axiom) R->L }
% 52.47/8.08 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 52.47/8.08 = { by axiom 5 (cn_1) R->L }
% 52.47/8.08 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x))))), implies(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))))), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 52.47/8.09 = { by axiom 4 (condensed_detachment) }
% 52.47/8.09 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 52.47/8.09 = { by axiom 1 (ifeq_axiom) }
% 52.47/8.10 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.15/8.10 = { by axiom 1 (ifeq_axiom) R->L }
% 53.15/8.10 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.15/8.11 = { by axiom 4 (condensed_detachment) R->L }
% 53.15/8.11 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x)))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.15/8.12 = { by axiom 1 (ifeq_axiom) R->L }
% 53.15/8.13 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.15/8.13 = { by axiom 5 (cn_1) R->L }
% 53.15/8.14 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x)))), implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.15/8.14 = { by axiom 1 (ifeq_axiom) R->L }
% 53.15/8.14 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x)))), implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.15/8.15 = { by axiom 2 (cn_3) R->L }
% 53.15/8.15 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x)))), implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))))), true, ifeq(is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))))), true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.15/8.16 = { by axiom 4 (condensed_detachment) }
% 53.15/8.16 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.15/8.17 = { by axiom 1 (ifeq_axiom) }
% 53.15/8.17 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.15/8.17 = { by axiom 1 (ifeq_axiom) R->L }
% 53.15/8.18 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true), true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.15/8.19 = { by axiom 5 (cn_1) R->L }
% 53.15/8.19 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true), true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.15/8.19 = { by axiom 1 (ifeq_axiom) R->L }
% 53.86/8.20 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x)))))), true, ifeq(true, true, is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true), true), true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.86/8.20 = { by axiom 3 (cn_2) R->L }
% 53.86/8.21 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x)))))), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(not(x))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true), true), true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.86/8.21 = { by axiom 4 (condensed_detachment) }
% 53.86/8.21 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.86/8.22 = { by axiom 1 (ifeq_axiom) }
% 53.86/8.22 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.86/8.22 = { by axiom 1 (ifeq_axiom) R->L }
% 53.86/8.22 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.86/8.23 = { by axiom 5 (cn_1) R->L }
% 53.86/8.23 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x)))), implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true, ifeq(is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))), not(not(x))), implies(not(not(not(x))), not(not(x)))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.86/8.23 = { by axiom 4 (condensed_detachment) }
% 53.86/8.23 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.86/8.24 = { by axiom 1 (ifeq_axiom) }
% 53.86/8.24 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.86/8.24 = { by axiom 1 (ifeq_axiom) R->L }
% 53.86/8.25 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.86/8.25 = { by axiom 4 (condensed_detachment) R->L }
% 53.86/8.25 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))))), true, ifeq(is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.86/8.26 = { by axiom 5 (cn_1) }
% 53.86/8.26 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.86/8.27 = { by axiom 1 (ifeq_axiom) }
% 53.86/8.27 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.86/8.28 = { by axiom 1 (ifeq_axiom) R->L }
% 53.86/8.28 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.86/8.28 = { by axiom 4 (condensed_detachment) R->L }
% 53.86/8.29 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))))), true, ifeq(is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 53.86/8.30 = { by axiom 5 (cn_1) }
% 54.56/8.30 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 54.56/8.31 = { by axiom 1 (ifeq_axiom) }
% 54.56/8.31 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 54.56/8.32 = { by axiom 1 (ifeq_axiom) R->L }
% 54.56/8.32 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 54.56/8.33 = { by axiom 5 (cn_1) R->L }
% 54.56/8.34 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 54.56/8.35 = { by axiom 1 (ifeq_axiom) R->L }
% 54.56/8.35 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 54.56/8.36 = { by axiom 4 (condensed_detachment) R->L }
% 54.56/8.37 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x)))))), implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))))), true, ifeq(is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 54.56/8.38 = { by axiom 2 (cn_3) }
% 54.56/8.38 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x)))))), implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 54.56/8.39 = { by axiom 1 (ifeq_axiom) }
% 54.56/8.40 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x)))))), implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))))), true, is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 55.27/8.40 = { by axiom 5 (cn_1) }
% 55.27/8.41 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 55.27/8.41 = { by axiom 1 (ifeq_axiom) }
% 55.27/8.42 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 55.27/8.43 = { by axiom 4 (condensed_detachment) }
% 55.27/8.43 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 55.27/8.44 = { by axiom 1 (ifeq_axiom) }
% 55.27/8.45 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 55.27/8.46 = { by axiom 1 (ifeq_axiom) R->L }
% 55.27/8.46 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true, ifeq(true, true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 55.27/8.47 = { by axiom 4 (condensed_detachment) R->L }
% 55.27/8.47 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true, ifeq(is_a_theorem(implies(not(not(not(x))), implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))))), true, is_a_theorem(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 55.27/8.48 = { by axiom 2 (cn_3) }
% 55.27/8.49 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 55.27/8.49 = { by axiom 1 (ifeq_axiom) }
% 55.27/8.50 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true, is_a_theorem(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 55.27/8.50 = { by axiom 5 (cn_1) }
% 55.27/8.50 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 55.27/8.51 = { by axiom 1 (ifeq_axiom) }
% 55.27/8.51 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x)))))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(not(x)))), not(not(implies(not(not(not(x))), not(not(x)))))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 55.27/8.51 = { by axiom 4 (condensed_detachment) }
% 55.27/8.51 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 55.27/8.52 = { by axiom 1 (ifeq_axiom) }
% 55.27/8.52 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))))), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), implies(not(not(not(implies(not(not(not(x))), not(not(x)))))), not(not(implies(not(not(not(x))), not(not(x))))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 55.27/8.52 = { by axiom 4 (condensed_detachment) }
% 55.27/8.52 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 55.27/8.53 = { by axiom 1 (ifeq_axiom) }
% 55.27/8.53 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.53 = { by axiom 1 (ifeq_axiom) R->L }
% 56.22/8.53 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.54 = { by axiom 4 (condensed_detachment) R->L }
% 56.22/8.54 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.54 = { by axiom 5 (cn_1) }
% 56.22/8.54 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.55 = { by axiom 1 (ifeq_axiom) }
% 56.22/8.55 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.56 = { by axiom 1 (ifeq_axiom) R->L }
% 56.22/8.56 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.56 = { by axiom 5 (cn_1) R->L }
% 56.22/8.56 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x)))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.57 = { by axiom 1 (ifeq_axiom) R->L }
% 56.22/8.57 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x)))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.57 = { by axiom 3 (cn_2) R->L }
% 56.22/8.57 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x)))))))), true, ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.57 = { by axiom 4 (condensed_detachment) }
% 56.22/8.58 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.58 = { by axiom 1 (ifeq_axiom) }
% 56.22/8.58 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.58 = { by axiom 4 (condensed_detachment) }
% 56.22/8.58 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.59 = { by axiom 1 (ifeq_axiom) }
% 56.22/8.59 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.59 = { by axiom 1 (ifeq_axiom) R->L }
% 56.22/8.59 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.59 = { by axiom 4 (condensed_detachment) R->L }
% 56.22/8.59 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.59 = { by axiom 3 (cn_2) }
% 56.22/8.60 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))))), true, ifeq(true, true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.60 = { by axiom 1 (ifeq_axiom) }
% 56.22/8.60 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.60 = { by axiom 1 (ifeq_axiom) R->L }
% 56.22/8.60 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.61 = { by axiom 5 (cn_1) R->L }
% 56.22/8.61 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x)))))))), true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))))), true), true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.61 = { by axiom 1 (ifeq_axiom) R->L }
% 56.22/8.61 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x)))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.61 = { by axiom 2 (cn_3) R->L }
% 56.22/8.61 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x)))))))), true, ifeq(is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))), implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))))), true), true), true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.62 = { by axiom 4 (condensed_detachment) }
% 56.22/8.62 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.62 = { by axiom 1 (ifeq_axiom) }
% 56.22/8.62 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x))))), implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x))))))), true, ifeq(is_a_theorem(implies(not(implies(not(not(not(x))), not(not(x)))), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), not(implies(not(not(not(x))), not(not(x))))), not(implies(not(not(not(x))), not(not(x)))))), true), true), true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.62 = { by axiom 4 (condensed_detachment) }
% 56.22/8.62 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.63 = { by axiom 1 (ifeq_axiom) }
% 56.22/8.63 ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.63 = { by axiom 1 (ifeq_axiom) R->L }
% 56.22/8.63 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.22/8.63 = { by axiom 4 (condensed_detachment) R->L }
% 56.94/8.63 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))))), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true, is_a_theorem(implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true), true), true, ifeq(is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.63 = { by axiom 1 (ifeq_axiom) R->L }
% 56.94/8.63 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true), true, is_a_theorem(implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true), true), true, ifeq(is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.63 = { by axiom 5 (cn_1) R->L }
% 56.94/8.64 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), implies(not(implies(not(not(not(x))), not(not(x)))), x)), implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true), true, is_a_theorem(implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true), true), true, ifeq(is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.64 = { by axiom 1 (ifeq_axiom) R->L }
% 56.94/8.64 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), implies(not(implies(not(not(not(x))), not(not(x)))), x)), implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true), true), true, ifeq(is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.64 = { by axiom 2 (cn_3) R->L }
% 56.94/8.64 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(x))), implies(not(implies(not(not(not(x))), not(not(x)))), x)), implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))))), true, ifeq(is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(implies(not(not(not(x))), not(not(x)))), x))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true), true), true, ifeq(is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.64 = { by axiom 4 (condensed_detachment) }
% 56.94/8.64 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))))), true, ifeq(true, true, is_a_theorem(implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true), true), true, ifeq(is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.64 = { by axiom 1 (ifeq_axiom) }
% 56.94/8.64 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))))), true, is_a_theorem(implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true), true, ifeq(is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.65 = { by axiom 1 (ifeq_axiom) R->L }
% 56.94/8.65 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))))), true), true, is_a_theorem(implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true), true, ifeq(is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.65 = { by axiom 5 (cn_1) R->L }
% 56.94/8.65 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))))), true), true, is_a_theorem(implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true), true, ifeq(is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.65 = { by axiom 1 (ifeq_axiom) R->L }
% 56.94/8.65 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))))), true), true), true, is_a_theorem(implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true), true, ifeq(is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.65 = { by axiom 5 (cn_1) R->L }
% 56.94/8.65 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x))), implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))))), true, ifeq(is_a_theorem(implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(not(x))), not(not(x)))), x), implies(not(x), x)), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))))), true), true), true, is_a_theorem(implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true), true, ifeq(is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.65 = { by axiom 4 (condensed_detachment) }
% 56.94/8.65 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true), true, ifeq(is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.65 = { by axiom 1 (ifeq_axiom) }
% 56.94/8.65 ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(x), not(implies(not(not(not(x))), not(not(x))))), implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x)))), true, ifeq(is_a_theorem(implies(not(x), not(implies(not(not(not(x))), not(not(x)))))), true, is_a_theorem(implies(implies(not(not(not(x))), not(not(x))), implies(not(x), x))), true), true), true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.65 = { by axiom 4 (condensed_detachment) }
% 56.94/8.65 ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(x)), implies(not(x), x))), true), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.65 = { by axiom 1 (ifeq_axiom) }
% 56.94/8.66 ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.66 = { by axiom 1 (ifeq_axiom) R->L }
% 56.94/8.66 ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.66 = { by axiom 4 (condensed_detachment) R->L }
% 56.94/8.66 ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x)))), implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x)))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.66 = { by axiom 1 (ifeq_axiom) R->L }
% 56.94/8.66 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x)))), implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.66 = { by axiom 5 (cn_1) R->L }
% 56.94/8.66 ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x)))), implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.66 = { by axiom 1 (ifeq_axiom) R->L }
% 56.94/8.66 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x)))), implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.66 = { by axiom 5 (cn_1) R->L }
% 56.94/8.66 ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x)))), implies(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x)))), implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))))), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x)))), implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.66 = { by axiom 4 (condensed_detachment) }
% 56.94/8.66 ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.66 = { by axiom 1 (ifeq_axiom) }
% 56.94/8.66 ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.67 = { by axiom 1 (ifeq_axiom) R->L }
% 56.94/8.67 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.67 = { by axiom 4 (condensed_detachment) R->L }
% 56.94/8.67 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), x), x), implies(implies(x, not(x)), implies(implies(not(x), x), not(x))))), true, ifeq(is_a_theorem(implies(implies(not(x), x), x)), true, is_a_theorem(implies(implies(x, not(x)), implies(implies(not(x), x), not(x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.67 = { by axiom 3 (cn_2) }
% 56.94/8.67 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), x), x), implies(implies(x, not(x)), implies(implies(not(x), x), not(x))))), true, ifeq(true, true, is_a_theorem(implies(implies(x, not(x)), implies(implies(not(x), x), not(x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.67 = { by axiom 1 (ifeq_axiom) }
% 56.94/8.67 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), x), x), implies(implies(x, not(x)), implies(implies(not(x), x), not(x))))), true, is_a_theorem(implies(implies(x, not(x)), implies(implies(not(x), x), not(x)))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.67 = { by axiom 5 (cn_1) }
% 56.94/8.67 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(x, not(x)), implies(implies(not(x), x), not(x)))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.67 = { by axiom 1 (ifeq_axiom) }
% 56.94/8.67 ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(x, not(x)), implies(implies(not(x), x), not(x)))), true, is_a_theorem(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.67 = { by axiom 1 (ifeq_axiom) R->L }
% 56.94/8.67 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(implies(not(x), x), not(x)))), true, is_a_theorem(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.67 = { by axiom 5 (cn_1) R->L }
% 56.94/8.67 ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(x, not(x)), implies(implies(not(x), x), not(x))), implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x)))))), true, ifeq(is_a_theorem(implies(implies(x, not(x)), implies(implies(not(x), x), not(x)))), true, is_a_theorem(implies(implies(implies(implies(not(x), x), not(x)), implies(not(not(x)), not(x))), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.67 = { by axiom 4 (condensed_detachment) }
% 56.94/8.67 ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.67 = { by axiom 1 (ifeq_axiom) }
% 56.94/8.67 ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(x), x)), implies(implies(x, not(x)), implies(not(not(x)), not(x))))), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(x), x))), true, is_a_theorem(implies(implies(x, not(x)), implies(not(not(x)), not(x)))), true), true), true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.67 = { by axiom 4 (condensed_detachment) }
% 56.94/8.67 ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.67 = { by axiom 1 (ifeq_axiom) }
% 56.94/8.67 ifeq(is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true)
% 56.94/8.67 = { by axiom 1 (ifeq_axiom) R->L }
% 56.94/8.67 ifeq(is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true, ifeq(true, true, is_a_theorem(implies(implies(x, not(x)), not(x))), true), true)
% 56.94/8.67 = { by axiom 4 (condensed_detachment) R->L }
% 56.94/8.67 ifeq(is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(not(x), not(x)))), true, ifeq(is_a_theorem(implies(implies(not(not(x)), not(x)), not(x))), true, is_a_theorem(implies(not(x), not(x))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true), true)
% 56.94/8.67 = { by axiom 3 (cn_2) }
% 56.94/8.67 ifeq(is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(not(x), not(x)))), true, ifeq(true, true, is_a_theorem(implies(not(x), not(x))), true), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true), true)
% 56.94/8.67 = { by axiom 1 (ifeq_axiom) }
% 56.94/8.67 ifeq(is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(not(x), not(x)))), true, is_a_theorem(implies(not(x), not(x))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true), true)
% 56.94/8.68 = { by axiom 1 (ifeq_axiom) R->L }
% 56.94/8.68 ifeq(is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true, ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(not(x), not(x)))), true), true, is_a_theorem(implies(not(x), not(x))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true), true)
% 56.94/8.68 = { by axiom 5 (cn_1) R->L }
% 56.94/8.68 ifeq(is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(x), implies(not(not(x)), not(x))), implies(implies(implies(not(not(x)), not(x)), not(x)), implies(not(x), not(x))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(not(x), not(x)))), true), true, is_a_theorem(implies(not(x), not(x))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true), true)
% 56.94/8.68 = { by axiom 1 (ifeq_axiom) R->L }
% 56.94/8.68 ifeq(is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(x), implies(not(not(x)), not(x))), implies(implies(implies(not(not(x)), not(x)), not(x)), implies(not(x), not(x))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(not(x), not(x)))), true), true), true, is_a_theorem(implies(not(x), not(x))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true), true)
% 56.94/8.68 = { by axiom 2 (cn_3) R->L }
% 56.94/8.68 ifeq(is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(x), implies(not(not(x)), not(x))), implies(implies(implies(not(not(x)), not(x)), not(x)), implies(not(x), not(x))))), true, ifeq(is_a_theorem(implies(not(x), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(not(x), not(x)))), true), true), true, is_a_theorem(implies(not(x), not(x))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true), true)
% 56.94/8.68 = { by axiom 4 (condensed_detachment) }
% 56.94/8.68 ifeq(is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true, ifeq(ifeq(true, true, is_a_theorem(implies(not(x), not(x))), true), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true), true)
% 56.94/8.68 = { by axiom 1 (ifeq_axiom) }
% 56.94/8.68 ifeq(is_a_theorem(implies(implies(not(x), not(x)), implies(implies(x, not(x)), not(x)))), true, ifeq(is_a_theorem(implies(not(x), not(x))), true, is_a_theorem(implies(implies(x, not(x)), not(x))), true), true)
% 56.94/8.68 = { by axiom 4 (condensed_detachment) }
% 56.94/8.68 true
% 56.94/8.68 % SZS output end Proof
% 56.94/8.68
% 56.94/8.68 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------