Twee---2.7.UNS-Prf.s

%------------------------------------------------------------------------------
% File     : Twee---2.7
% Problem  : LCL052-1 : TPTP v9.3.1. Released v1.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p

% Computer : n008.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 11:48:55 AM UTC 2026

% Result   : Unsatisfiable 36.23s 5.01s
% Output   : Proof 55.22s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02  % Problem  : LCL052-1 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.04  % Command  : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.38  % Computer : n008.cluster.edu
% 0.09/0.38  % Model    : x86_64 x86_64
% 0.09/0.38  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.38  % Memory   : 8046.5625MB
% 0.09/0.38  % OS       : Linux 6.8.0-71-generic
% 0.09/0.38  % CPULimit : 300
% 0.09/0.38  % WCLimit  : 300
% 0.09/0.38  % DateTime : Sun Sep 27 15:19:39 UTC 2026
% 0.09/0.38  % CPUTime  : 
% 0.09/0.38  Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 36.23/5.01  Command-line arguments: --no-flatten-goal
% 36.23/5.01  
% 36.23/5.01  % SZS status Unsatisfiable
% 36.23/5.01  
% 49.71/6.75  % SZS output start Proof
% 49.71/6.75  Axiom 1 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 49.71/6.75  Axiom 2 (cn_3): is_a_theorem(implies(X, implies(not(X), Y))) = true.
% 49.71/6.75  Axiom 3 (cn_2): is_a_theorem(implies(implies(not(X), X), X)) = true.
% 49.71/6.75  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.
% 49.71/6.78  Axiom 5 (cn_1): is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
% 49.71/6.78  
% 49.71/6.78  Goal 1 (prove_cn_24): is_a_theorem(implies(implies(implies(a, b), a), a)) = true.
% 49.71/6.78  Proof:
% 49.71/6.78    is_a_theorem(implies(implies(implies(a, b), a), a))
% 49.71/6.78  = { by axiom 1 (ifeq_axiom) R->L }
% 49.71/6.78    ifeq(true, true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 49.71/6.78  = { by axiom 4 (condensed_detachment) R->L }
% 49.71/6.78    ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 49.71/6.78  = { by axiom 1 (ifeq_axiom) R->L }
% 49.71/6.78    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 49.71/6.78  = { by axiom 4 (condensed_detachment) R->L }
% 49.71/6.78    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(a, a), implies(implies(not(a), a), a)), implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a))))), true, ifeq(is_a_theorem(implies(implies(a, a), implies(implies(not(a), a), a))), true, is_a_theorem(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 49.71/6.78  = { by axiom 5 (cn_1) }
% 49.71/6.78    ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(a, a), implies(implies(not(a), a), a))), true, is_a_theorem(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 49.71/6.79  = { by axiom 1 (ifeq_axiom) }
% 49.71/6.79    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, a), implies(implies(not(a), a), a))), true, is_a_theorem(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 49.71/6.79  = { by axiom 1 (ifeq_axiom) R->L }
% 49.71/6.79    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(a, a), implies(implies(not(a), a), a))), true), true, is_a_theorem(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 49.71/6.79  = { by axiom 5 (cn_1) R->L }
% 49.71/6.79    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(a), a), a), implies(implies(a, a), implies(implies(not(a), a), a)))), true, is_a_theorem(implies(implies(a, a), implies(implies(not(a), a), a))), true), true, is_a_theorem(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 49.71/6.79  = { by axiom 1 (ifeq_axiom) R->L }
% 49.71/6.79    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(a), a), a), implies(implies(a, a), implies(implies(not(a), a), a)))), true, ifeq(true, true, is_a_theorem(implies(implies(a, a), implies(implies(not(a), a), a))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 49.71/6.79  = { by axiom 3 (cn_2) R->L }
% 49.71/6.79    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(a), a), a), implies(implies(a, a), implies(implies(not(a), a), a)))), true, ifeq(is_a_theorem(implies(implies(not(a), a), a)), true, is_a_theorem(implies(implies(a, a), implies(implies(not(a), a), a))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 49.71/6.79  = { by axiom 4 (condensed_detachment) }
% 49.71/6.79    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 49.71/6.79  = { by axiom 1 (ifeq_axiom) }
% 49.71/6.79    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true, is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 49.71/6.79  = { by axiom 1 (ifeq_axiom) R->L }
% 49.71/6.79    ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true, is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 49.71/6.80  = { by axiom 4 (condensed_detachment) R->L }
% 49.71/6.80    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a))), implies(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a))), implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a))), implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true, is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 49.71/6.80  = { by axiom 5 (cn_1) }
% 49.71/6.80    ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a))), implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true, is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 49.71/6.80  = { by axiom 1 (ifeq_axiom) }
% 50.53/6.80    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a))), implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true, is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.80  = { by axiom 5 (cn_1) }
% 50.53/6.80    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a))), implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true, is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.81  = { by axiom 1 (ifeq_axiom) }
% 50.53/6.81    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a))), implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true, is_a_theorem(implies(implies(implies(implies(a, b), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.81  = { by axiom 4 (condensed_detachment) }
% 50.53/6.81    ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.81  = { by axiom 1 (ifeq_axiom) }
% 50.53/6.81    ifeq(ifeq(is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.81  = { by axiom 1 (ifeq_axiom) R->L }
% 50.53/6.81    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.81  = { by axiom 4 (condensed_detachment) R->L }
% 50.53/6.81    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.81  = { by axiom 1 (ifeq_axiom) R->L }
% 50.53/6.81    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.81  = { by axiom 4 (condensed_detachment) R->L }
% 50.53/6.81    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b))), implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))))), true, ifeq(is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true), true), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.81  = { by axiom 5 (cn_1) }
% 50.53/6.81    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true), true), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.81  = { by axiom 1 (ifeq_axiom) }
% 50.53/6.81    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.81  = { by axiom 5 (cn_1) }
% 50.53/6.82    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.82  = { by axiom 1 (ifeq_axiom) }
% 50.53/6.82    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.82  = { by axiom 1 (ifeq_axiom) R->L }
% 50.53/6.82    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, ifeq(true, true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.82  = { by axiom 4 (condensed_detachment) R->L }
% 50.53/6.82    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(not(implies(not(not(a)), not(a))), b)), implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(not(implies(not(not(a)), not(a))), b))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.82  = { by axiom 2 (cn_3) }
% 50.53/6.82    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(not(implies(not(not(a)), not(a))), b)), implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.82  = { by axiom 1 (ifeq_axiom) }
% 50.53/6.82    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(not(implies(not(not(a)), not(a))), b)), implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.83  = { by axiom 5 (cn_1) }
% 50.53/6.83    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.83  = { by axiom 1 (ifeq_axiom) }
% 50.53/6.83    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b)))), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.83  = { by axiom 4 (condensed_detachment) }
% 50.53/6.83    ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.83  = { by axiom 1 (ifeq_axiom) }
% 50.53/6.83    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.83  = { by axiom 1 (ifeq_axiom) R->L }
% 50.53/6.83    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.83  = { by axiom 4 (condensed_detachment) R->L }
% 50.53/6.83    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.84  = { by axiom 1 (ifeq_axiom) R->L }
% 50.53/6.84    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.84  = { by axiom 4 (condensed_detachment) R->L }
% 50.53/6.84    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.84  = { by axiom 2 (cn_3) }
% 50.53/6.84    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.85  = { by axiom 1 (ifeq_axiom) }
% 50.53/6.85    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.85  = { by axiom 5 (cn_1) }
% 50.53/6.85    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.85  = { by axiom 1 (ifeq_axiom) }
% 50.53/6.85    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.85  = { by axiom 1 (ifeq_axiom) R->L }
% 50.53/6.85    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true, ifeq(true, true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.85  = { by axiom 3 (cn_2) R->L }
% 50.53/6.85    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.86  = { by axiom 4 (condensed_detachment) }
% 50.53/6.86    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.86  = { by axiom 1 (ifeq_axiom) }
% 50.53/6.86    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.86  = { by axiom 1 (ifeq_axiom) R->L }
% 50.53/6.86    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.86  = { by axiom 4 (condensed_detachment) R->L }
% 50.53/6.87    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.87  = { by axiom 1 (ifeq_axiom) R->L }
% 50.53/6.87    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.87  = { by axiom 4 (condensed_detachment) R->L }
% 50.53/6.88    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.88  = { by axiom 3 (cn_2) }
% 50.53/6.88    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.88  = { by axiom 1 (ifeq_axiom) }
% 50.53/6.88    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.88  = { by axiom 5 (cn_1) }
% 50.53/6.89    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.89  = { by axiom 1 (ifeq_axiom) }
% 50.53/6.89    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.89  = { by axiom 1 (ifeq_axiom) R->L }
% 50.53/6.89    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.89  = { by axiom 5 (cn_1) R->L }
% 50.53/6.90    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 50.53/6.90  = { by axiom 4 (condensed_detachment) }
% 50.53/6.90    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 51.29/6.90  = { by axiom 1 (ifeq_axiom) }
% 51.29/6.90    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 51.29/6.90  = { by axiom 1 (ifeq_axiom) R->L }
% 51.29/6.91    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 51.29/6.91  = { by axiom 4 (condensed_detachment) R->L }
% 51.29/6.91    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 51.29/6.92  = { by axiom 1 (ifeq_axiom) R->L }
% 51.29/6.92    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 51.29/6.93  = { by axiom 4 (condensed_detachment) R->L }
% 51.29/6.93    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 51.29/6.94  = { by axiom 1 (ifeq_axiom) R->L }
% 51.29/6.94    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 51.29/6.95  = { by axiom 5 (cn_1) R->L }
% 51.29/6.95    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(a))), not(not(implies(not(not(a)), not(a)))))), implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, is_a_theorem(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 51.29/6.95  = { by axiom 1 (ifeq_axiom) R->L }
% 51.29/6.96    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(a))), not(not(implies(not(not(a)), not(a)))))), implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 51.29/6.96  = { by axiom 2 (cn_3) R->L }
% 51.29/6.97    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(a))), not(not(implies(not(not(a)), not(a)))))), implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(is_a_theorem(implies(not(not(a)), implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 51.29/6.97  = { by axiom 4 (condensed_detachment) }
% 51.29/6.97    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(true, true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 51.29/6.98  = { by axiom 1 (ifeq_axiom) }
% 51.29/6.98    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 51.29/6.99  = { by axiom 1 (ifeq_axiom) R->L }
% 51.29/6.99    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(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 51.29/7.00  = { by axiom 4 (condensed_detachment) R->L }
% 52.11/7.00    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(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.01  = { by axiom 1 (ifeq_axiom) R->L }
% 52.11/7.02    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(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.02  = { by axiom 5 (cn_1) R->L }
% 52.11/7.03    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(a)), not(a)))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a))))), implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))))), true, is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.04  = { by axiom 1 (ifeq_axiom) R->L }
% 52.11/7.04    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(a)), not(a)))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a))))), implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.04  = { by axiom 2 (cn_3) R->L }
% 52.11/7.05    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(a)), not(a)))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a))))), implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))))), true, ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.05  = { by axiom 4 (condensed_detachment) }
% 52.11/7.05    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(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.06  = { by axiom 1 (ifeq_axiom) }
% 52.11/7.06    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(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.07  = { by axiom 5 (cn_1) }
% 52.11/7.07    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(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.08  = { by axiom 1 (ifeq_axiom) }
% 52.11/7.08    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.08  = { by axiom 1 (ifeq_axiom) R->L }
% 52.11/7.08    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(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.09  = { by axiom 5 (cn_1) R->L }
% 52.11/7.10    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))))), true, ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.10  = { by axiom 4 (condensed_detachment) }
% 52.11/7.10    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.10  = { by axiom 1 (ifeq_axiom) }
% 52.11/7.10    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.11  = { by axiom 1 (ifeq_axiom) R->L }
% 52.11/7.11    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.11  = { by axiom 5 (cn_1) R->L }
% 52.11/7.11    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.12  = { by axiom 4 (condensed_detachment) }
% 52.11/7.12    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.12  = { by axiom 1 (ifeq_axiom) }
% 52.11/7.12    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.12  = { by axiom 1 (ifeq_axiom) R->L }
% 52.11/7.12    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.13  = { by axiom 4 (condensed_detachment) R->L }
% 52.11/7.13    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true, ifeq(is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.13  = { by axiom 5 (cn_1) }
% 52.11/7.13    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.14  = { by axiom 1 (ifeq_axiom) }
% 52.11/7.14    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.14  = { by axiom 1 (ifeq_axiom) R->L }
% 52.11/7.15    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.15  = { by axiom 4 (condensed_detachment) R->L }
% 52.11/7.15    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(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))))), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.16  = { by axiom 3 (cn_2) }
% 52.11/7.16    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(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))))), true, ifeq(true, true, is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.16  = { by axiom 1 (ifeq_axiom) }
% 52.11/7.16    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(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))))), true, is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 52.11/7.17  = { by axiom 5 (cn_1) }
% 53.33/7.17    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(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.17  = { by axiom 1 (ifeq_axiom) }
% 53.33/7.18    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.18  = { by axiom 1 (ifeq_axiom) R->L }
% 53.33/7.18    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(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.19  = { by axiom 4 (condensed_detachment) R->L }
% 53.33/7.19    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(implies(not(not(a)), not(a)))), not(a))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))))), true, ifeq(is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(not(not(implies(not(not(a)), not(a)))), not(a)))), true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.19  = { by axiom 2 (cn_3) }
% 53.33/7.20    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(implies(not(not(a)), not(a)))), not(a))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.20  = { by axiom 1 (ifeq_axiom) }
% 53.33/7.20    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(implies(not(not(a)), not(a)))), not(a))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.21  = { by axiom 5 (cn_1) }
% 53.33/7.21    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(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.21  = { by axiom 1 (ifeq_axiom) }
% 53.33/7.21    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.22  = { by axiom 4 (condensed_detachment) }
% 53.33/7.22    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(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.22  = { by axiom 1 (ifeq_axiom) }
% 53.33/7.23    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.23  = { by axiom 1 (ifeq_axiom) R->L }
% 53.33/7.23    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(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.23  = { by axiom 4 (condensed_detachment) R->L }
% 53.33/7.24    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a)))), implies(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.24  = { by axiom 5 (cn_1) }
% 53.33/7.24    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.25  = { by axiom 1 (ifeq_axiom) }
% 53.33/7.25    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.25  = { by axiom 5 (cn_1) }
% 53.33/7.25    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(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.25  = { by axiom 1 (ifeq_axiom) }
% 53.33/7.26    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(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.26  = { by axiom 4 (condensed_detachment) }
% 53.33/7.26    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.26  = { by axiom 1 (ifeq_axiom) }
% 53.33/7.26    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.26  = { by axiom 1 (ifeq_axiom) R->L }
% 53.33/7.26    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 53.33/7.27  = { by axiom 5 (cn_1) R->L }
% 53.33/7.27    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.18/7.27  = { by axiom 4 (condensed_detachment) }
% 54.18/7.27    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.18/7.27  = { by axiom 1 (ifeq_axiom) }
% 54.18/7.27    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.18/7.28  = { by axiom 1 (ifeq_axiom) R->L }
% 54.18/7.28    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.18/7.28  = { by axiom 4 (condensed_detachment) R->L }
% 54.18/7.29    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.18/7.29  = { by axiom 1 (ifeq_axiom) R->L }
% 54.18/7.30    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.18/7.30  = { by axiom 5 (cn_1) R->L }
% 54.48/7.30    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.31  = { by axiom 1 (ifeq_axiom) R->L }
% 54.48/7.31    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.31  = { by axiom 5 (cn_1) R->L }
% 54.48/7.32    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), implies(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.32  = { by axiom 4 (condensed_detachment) }
% 54.48/7.32    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.33  = { by axiom 1 (ifeq_axiom) }
% 54.48/7.33    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.33  = { by axiom 1 (ifeq_axiom) R->L }
% 54.48/7.33    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.34  = { by axiom 4 (condensed_detachment) R->L }
% 54.48/7.34    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.34  = { by axiom 3 (cn_2) }
% 54.48/7.34    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, ifeq(true, true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.35  = { by axiom 1 (ifeq_axiom) }
% 54.48/7.35    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.35  = { by axiom 5 (cn_1) }
% 54.48/7.35    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.36  = { by axiom 1 (ifeq_axiom) }
% 54.48/7.36    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.36  = { by axiom 1 (ifeq_axiom) R->L }
% 54.48/7.36    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.36  = { by axiom 5 (cn_1) R->L }
% 54.48/7.37    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))))), true, ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.37  = { by axiom 4 (condensed_detachment) }
% 54.48/7.37    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.37  = { by axiom 1 (ifeq_axiom) }
% 54.48/7.37    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.37  = { by axiom 4 (condensed_detachment) }
% 54.48/7.37    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.37  = { by axiom 1 (ifeq_axiom) }
% 54.48/7.37    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.37  = { by axiom 1 (ifeq_axiom) R->L }
% 54.48/7.37    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.37  = { by axiom 4 (condensed_detachment) R->L }
% 54.48/7.37    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.38  = { by axiom 5 (cn_1) }
% 54.48/7.38    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.38  = { by axiom 1 (ifeq_axiom) }
% 54.48/7.38    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.38  = { by axiom 1 (ifeq_axiom) R->L }
% 54.48/7.38    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.38  = { by axiom 4 (condensed_detachment) R->L }
% 54.48/7.38    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(a)), implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), not(a))), true, is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.38  = { by axiom 3 (cn_2) }
% 54.48/7.38    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(a)), implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))))), true, ifeq(true, true, is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.39  = { by axiom 1 (ifeq_axiom) }
% 54.48/7.39    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(a)), implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.39  = { by axiom 5 (cn_1) }
% 54.48/7.39    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.39  = { by axiom 1 (ifeq_axiom) }
% 54.48/7.39    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.39  = { by axiom 1 (ifeq_axiom) R->L }
% 54.48/7.39    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 54.48/7.40  = { by axiom 4 (condensed_detachment) R->L }
% 55.22/7.40    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, implies(not(a), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))))), true, ifeq(is_a_theorem(implies(a, implies(not(a), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.40  = { by axiom 2 (cn_3) }
% 55.22/7.40    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, implies(not(a), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.40  = { by axiom 1 (ifeq_axiom) }
% 55.22/7.40    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, implies(not(a), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.40  = { by axiom 5 (cn_1) }
% 55.22/7.41    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.41  = { by axiom 1 (ifeq_axiom) }
% 55.22/7.41    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.41  = { by axiom 4 (condensed_detachment) }
% 55.22/7.41    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.41  = { by axiom 1 (ifeq_axiom) }
% 55.22/7.41    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.41  = { by axiom 4 (condensed_detachment) }
% 55.22/7.41    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.41  = { by axiom 1 (ifeq_axiom) }
% 55.22/7.41    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.41  = { by axiom 1 (ifeq_axiom) R->L }
% 55.22/7.41    ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.41  = { by axiom 4 (condensed_detachment) R->L }
% 55.22/7.41    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))))), true, ifeq(is_a_theorem(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a)))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.42  = { by axiom 5 (cn_1) }
% 55.22/7.42    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.42  = { by axiom 1 (ifeq_axiom) }
% 55.22/7.42    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.42  = { by axiom 1 (ifeq_axiom) R->L }
% 55.22/7.42    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.42  = { by axiom 4 (condensed_detachment) R->L }
% 55.22/7.42    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(a), implies(not(not(a)), not(a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b))))), true, ifeq(is_a_theorem(implies(not(a), implies(not(not(a)), not(a)))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b)))), true), true), true, is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.42  = { by axiom 2 (cn_3) }
% 55.22/7.42    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(a), implies(not(not(a)), not(a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b)))), true), true), true, is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.42  = { by axiom 1 (ifeq_axiom) }
% 55.22/7.42    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(a), implies(not(not(a)), not(a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b)))), true), true, is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.42  = { by axiom 5 (cn_1) }
% 55.22/7.42    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b)))), true), true, is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.42  = { by axiom 1 (ifeq_axiom) }
% 55.22/7.42    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b)))), true, is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.42  = { by axiom 1 (ifeq_axiom) R->L }
% 55.22/7.42    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b)))), true, is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.43  = { by axiom 5 (cn_1) R->L }
% 55.22/7.43    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b))), implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a)))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b)))), true, is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.43  = { by axiom 4 (condensed_detachment) }
% 55.22/7.43    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.43  = { by axiom 1 (ifeq_axiom) }
% 55.22/7.43    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), a), implies(not(a), a)))), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), a), implies(not(a), a))), true), true), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.43  = { by axiom 4 (condensed_detachment) }
% 55.22/7.43    ifeq(ifeq(true, true, is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.43  = { by axiom 1 (ifeq_axiom) }
% 55.22/7.43    ifeq(is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 55.22/7.43  = { by axiom 1 (ifeq_axiom) R->L }
% 55.22/7.43    ifeq(is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 55.22/7.43  = { by axiom 4 (condensed_detachment) R->L }
% 55.22/7.43    ifeq(is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(a), a), a), implies(a, a))), true, ifeq(is_a_theorem(implies(implies(not(a), a), a)), true, is_a_theorem(implies(a, a)), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 55.22/7.43  = { by axiom 3 (cn_2) }
% 55.22/7.43    ifeq(is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(a), a), a), implies(a, a))), true, ifeq(true, true, is_a_theorem(implies(a, a)), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 55.22/7.43  = { by axiom 1 (ifeq_axiom) }
% 55.22/7.43    ifeq(is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(a), a), a), implies(a, a))), true, is_a_theorem(implies(a, a)), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 55.22/7.43  = { by axiom 1 (ifeq_axiom) R->L }
% 55.22/7.43    ifeq(is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(a), a), a), implies(a, a))), true), true, is_a_theorem(implies(a, a)), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 55.22/7.43  = { by axiom 5 (cn_1) R->L }
% 55.22/7.43    ifeq(is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, implies(not(a), a)), implies(implies(implies(not(a), a), a), implies(a, a)))), true, is_a_theorem(implies(implies(implies(not(a), a), a), implies(a, a))), true), true, is_a_theorem(implies(a, a)), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 55.22/7.43  = { by axiom 1 (ifeq_axiom) R->L }
% 55.22/7.43    ifeq(is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, implies(not(a), a)), implies(implies(implies(not(a), a), a), implies(a, a)))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(a), a), a), implies(a, a))), true), true), true, is_a_theorem(implies(a, a)), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 55.22/7.43  = { by axiom 2 (cn_3) R->L }
% 55.22/7.43    ifeq(is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, implies(not(a), a)), implies(implies(implies(not(a), a), a), implies(a, a)))), true, ifeq(is_a_theorem(implies(a, implies(not(a), a))), true, is_a_theorem(implies(implies(implies(not(a), a), a), implies(a, a))), true), true), true, is_a_theorem(implies(a, a)), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 55.22/7.43  = { by axiom 4 (condensed_detachment) }
% 55.22/7.43    ifeq(is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(true, true, is_a_theorem(implies(a, a)), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 55.22/7.43  = { by axiom 1 (ifeq_axiom) }
% 55.22/7.43    ifeq(is_a_theorem(implies(implies(a, a), implies(implies(implies(a, b), a), a))), true, ifeq(is_a_theorem(implies(a, a)), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 55.22/7.43  = { by axiom 4 (condensed_detachment) }
% 55.22/7.43    true
% 55.22/7.43  % SZS output end Proof
% 55.22/7.43  
% 55.22/7.43  RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------