Twee---2.7.UNS-Prf.s

%------------------------------------------------------------------------------
% File     : Twee---2.7
% Problem  : LCL378-1 : TPTP v9.3.1. Released v2.3.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p

% Computer : n005.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8046.5625MB
% OS       : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Tue Sep 29 11:49:42 AM UTC 2026

% Result   : Unsatisfiable 20.50s 3.03s
% Output   : Proof 32.44s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03  % Problem  : LCL378-1 : TPTP v9.3.1. Released v2.3.0.
% 0.00/0.04  % Command  : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.08/0.35  % Computer : n005.cluster.edu
% 0.08/0.35  % Model    : x86_64 x86_64
% 0.08/0.35  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.35  % Memory   : 8046.5625MB
% 0.08/0.35  % OS       : Linux 6.8.0-71-generic
% 0.08/0.35  % CPULimit : 300
% 0.08/0.35  % WCLimit  : 300
% 0.08/0.35  % DateTime : Sun Sep 27 15:38:01 UTC 2026
% 0.08/0.36  % CPUTime  : 
% 0.08/0.36  Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 20.50/3.03  Command-line arguments: --no-flatten-goal
% 20.50/3.03  
% 20.50/3.03  % SZS status Unsatisfiable
% 20.50/3.03  
% 28.44/4.03  % SZS output start Proof
% 28.44/4.03  Axiom 1 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 28.44/4.03  Axiom 2 (cn_3): is_a_theorem(implies(X, implies(not(X), Y))) = true.
% 28.44/4.03  Axiom 3 (cn_2): is_a_theorem(implies(implies(not(X), X), X)) = true.
% 28.44/4.03  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.
% 28.44/4.06  Axiom 5 (cn_1): is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
% 28.44/4.06  
% 28.44/4.06  Goal 1 (prove_cn_36): is_a_theorem(implies(not(x), implies(x, y))) = true.
% 28.44/4.06  Proof:
% 28.44/4.06    is_a_theorem(implies(not(x), implies(x, y)))
% 28.44/4.06  = { by axiom 1 (ifeq_axiom) R->L }
% 28.44/4.06    ifeq(true, true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.06  = { by axiom 4 (condensed_detachment) R->L }
% 28.44/4.07    ifeq(ifeq(is_a_theorem(implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y)))), true, ifeq(is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.07  = { by axiom 1 (ifeq_axiom) R->L }
% 28.44/4.07    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y)))), true), true, ifeq(is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.07  = { by axiom 4 (condensed_detachment) R->L }
% 28.44/4.07    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y))), implies(implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y))), implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y)))))), true, ifeq(is_a_theorem(implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y))), implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y))))), true), true), true, is_a_theorem(implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y)))), true), true, ifeq(is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.07  = { by axiom 5 (cn_1) }
% 28.44/4.07    ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y))), implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y))))), true), true), true, is_a_theorem(implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y)))), true), true, ifeq(is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.07  = { by axiom 1 (ifeq_axiom) }
% 28.44/4.07    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y))), implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y))))), true), true, is_a_theorem(implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y)))), true), true, ifeq(is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.07  = { by axiom 5 (cn_1) }
% 28.44/4.07    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y))), implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y))))), true), true, is_a_theorem(implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y)))), true), true, ifeq(is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.07  = { by axiom 1 (ifeq_axiom) }
% 28.44/4.08    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y))), implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y))))), true, is_a_theorem(implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y)))), true), true, ifeq(is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.08  = { by axiom 1 (ifeq_axiom) R->L }
% 28.44/4.08    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y))), implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y))))), true, ifeq(true, true, is_a_theorem(implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y)))), true), true), true, ifeq(is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.08  = { by axiom 4 (condensed_detachment) R->L }
% 28.44/4.08    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y))), implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), implies(not(implies(not(not(x)), not(x))), y)), implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y))))), true, ifeq(is_a_theorem(implies(implies(not(not(x)), not(x)), implies(not(implies(not(not(x)), not(x))), y))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y)))), true), true), true, is_a_theorem(implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y)))), true), true), true, ifeq(is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.08  = { by axiom 2 (cn_3) }
% 28.44/4.08    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y))), implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), implies(not(implies(not(not(x)), not(x))), y)), implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y)))), true), true), true, is_a_theorem(implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y)))), true), true), true, ifeq(is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.08  = { by axiom 1 (ifeq_axiom) }
% 28.44/4.08    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y))), implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), implies(not(implies(not(not(x)), not(x))), y)), implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y))))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y)))), true), true, is_a_theorem(implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y)))), true), true), true, ifeq(is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.08  = { by axiom 5 (cn_1) }
% 28.44/4.08    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y))), implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y)))), true), true, is_a_theorem(implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y)))), true), true), true, ifeq(is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.08  = { by axiom 1 (ifeq_axiom) }
% 28.44/4.08    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y))), implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y))))), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), y), implies(x, y)), implies(implies(not(not(x)), not(x)), implies(x, y)))), true, is_a_theorem(implies(implies(x, not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), implies(x, y)))), true), true), true, ifeq(is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.08  = { by axiom 4 (condensed_detachment) }
% 28.44/4.08    ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.08  = { by axiom 1 (ifeq_axiom) }
% 28.44/4.08    ifeq(ifeq(is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.08  = { by axiom 1 (ifeq_axiom) R->L }
% 28.44/4.08    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.08  = { by axiom 4 (condensed_detachment) R->L }
% 28.44/4.08    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, ifeq(is_a_theorem(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.09  = { by axiom 1 (ifeq_axiom) R->L }
% 28.44/4.09    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.09  = { by axiom 4 (condensed_detachment) R->L }
% 28.44/4.09    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x))))))), true, ifeq(is_a_theorem(implies(not(implies(not(not(x)), not(x))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.09  = { by axiom 2 (cn_3) }
% 28.44/4.09    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.09  = { by axiom 1 (ifeq_axiom) }
% 28.44/4.09    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.09  = { by axiom 5 (cn_1) }
% 28.44/4.09    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.10  = { by axiom 1 (ifeq_axiom) }
% 28.44/4.10    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.10  = { by axiom 1 (ifeq_axiom) R->L }
% 28.44/4.10    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))))), true, ifeq(true, true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.10  = { by axiom 3 (cn_2) R->L }
% 28.44/4.10    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 28.44/4.10  = { by axiom 4 (condensed_detachment) }
% 28.44/4.10    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.10  = { by axiom 1 (ifeq_axiom) }
% 29.19/4.10    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.10  = { by axiom 1 (ifeq_axiom) R->L }
% 29.19/4.10    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.11  = { by axiom 4 (condensed_detachment) R->L }
% 29.19/4.11    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.11  = { by axiom 1 (ifeq_axiom) R->L }
% 29.19/4.11    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.11  = { by axiom 4 (condensed_detachment) R->L }
% 29.19/4.12    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x))))))), true, ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.12  = { by axiom 3 (cn_2) }
% 29.19/4.12    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.12  = { by axiom 1 (ifeq_axiom) }
% 29.19/4.12    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.13  = { by axiom 5 (cn_1) }
% 29.19/4.13    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.13  = { by axiom 1 (ifeq_axiom) }
% 29.19/4.13    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.13  = { by axiom 1 (ifeq_axiom) R->L }
% 29.19/4.13    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.13  = { by axiom 5 (cn_1) R->L }
% 29.19/4.14    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x))))), implies(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.14  = { by axiom 4 (condensed_detachment) }
% 29.19/4.14    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.14  = { by axiom 1 (ifeq_axiom) }
% 29.19/4.14    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.14  = { by axiom 1 (ifeq_axiom) R->L }
% 29.19/4.14    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.15  = { by axiom 4 (condensed_detachment) R->L }
% 29.19/4.15    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.15  = { by axiom 1 (ifeq_axiom) R->L }
% 29.19/4.16    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.16  = { by axiom 4 (condensed_detachment) R->L }
% 29.19/4.16    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.17  = { by axiom 1 (ifeq_axiom) R->L }
% 29.19/4.17    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.18  = { by axiom 5 (cn_1) R->L }
% 29.19/4.18    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(x))), not(not(implies(not(not(x)), not(x)))))), implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.18  = { by axiom 1 (ifeq_axiom) R->L }
% 29.19/4.19    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(x))), not(not(implies(not(not(x)), not(x)))))), implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.19  = { by axiom 2 (cn_3) R->L }
% 29.19/4.19    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(x))), not(not(implies(not(not(x)), not(x)))))), implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true, ifeq(is_a_theorem(implies(not(not(x)), implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))))), true, is_a_theorem(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 29.19/4.20  = { by axiom 4 (condensed_detachment) }
% 29.19/4.20    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true, ifeq(true, true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.05/4.21  = { by axiom 1 (ifeq_axiom) }
% 30.05/4.21    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.05/4.21  = { by axiom 1 (ifeq_axiom) R->L }
% 30.05/4.22    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.05/4.22  = { by axiom 4 (condensed_detachment) R->L }
% 30.05/4.23    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.05/4.23  = { by axiom 1 (ifeq_axiom) R->L }
% 30.05/4.24    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(x)), not(x))))), not(not(not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.05/4.24  = { by axiom 5 (cn_1) R->L }
% 30.05/4.25    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(not(x))))), implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))))), true, is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.05/4.26  = { by axiom 1 (ifeq_axiom) R->L }
% 30.05/4.26    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(not(x))))), implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.05/4.26  = { by axiom 2 (cn_3) R->L }
% 30.05/4.27    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(not(x))))), implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))))), true, ifeq(is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(not(x)))))), true, is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.05/4.28  = { by axiom 4 (condensed_detachment) }
% 30.05/4.28    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(x)), not(x))))), not(not(not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.05/4.28  = { by axiom 1 (ifeq_axiom) }
% 30.05/4.29    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.05/4.29  = { by axiom 5 (cn_1) }
% 30.05/4.29    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.05/4.30  = { by axiom 1 (ifeq_axiom) }
% 30.05/4.30    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.30  = { by axiom 1 (ifeq_axiom) R->L }
% 30.81/4.30    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.31  = { by axiom 5 (cn_1) R->L }
% 30.81/4.31    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))))), true, ifeq(is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(x))), not(not(implies(not(not(x)), not(x))))), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.31  = { by axiom 4 (condensed_detachment) }
% 30.81/4.31    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.32  = { by axiom 1 (ifeq_axiom) }
% 30.81/4.32    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.32  = { by axiom 1 (ifeq_axiom) R->L }
% 30.81/4.32    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.32  = { by axiom 5 (cn_1) R->L }
% 30.81/4.32    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))))), implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))))), true, ifeq(is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))))), true, is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.33  = { by axiom 4 (condensed_detachment) }
% 30.81/4.33    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.33  = { by axiom 1 (ifeq_axiom) }
% 30.81/4.33    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.33  = { by axiom 1 (ifeq_axiom) R->L }
% 30.81/4.33    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.34  = { by axiom 4 (condensed_detachment) R->L }
% 30.81/4.34    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x))), implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true, ifeq(is_a_theorem(implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.34  = { by axiom 5 (cn_1) }
% 30.81/4.34    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.35  = { by axiom 1 (ifeq_axiom) }
% 30.81/4.35    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.35  = { by axiom 1 (ifeq_axiom) R->L }
% 30.81/4.35    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.36  = { by axiom 4 (condensed_detachment) R->L }
% 30.81/4.36    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x))))), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true), true), true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.36  = { by axiom 3 (cn_2) }
% 30.81/4.36    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x))))), true, ifeq(true, true, is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true), true), true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.36  = { by axiom 1 (ifeq_axiom) }
% 30.81/4.37    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x))))), true, is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true), true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.37  = { by axiom 5 (cn_1) }
% 30.81/4.37    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true), true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.38  = { by axiom 1 (ifeq_axiom) }
% 30.81/4.38    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.38  = { by axiom 1 (ifeq_axiom) R->L }
% 30.81/4.38    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.39  = { by axiom 4 (condensed_detachment) R->L }
% 30.81/4.39    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(implies(not(not(x)), not(x)))), not(x))), implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x))), implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))))), true, ifeq(is_a_theorem(implies(not(implies(not(not(x)), not(x))), implies(not(not(implies(not(not(x)), not(x)))), not(x)))), true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x))), implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.39  = { by axiom 2 (cn_3) }
% 30.81/4.40    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(implies(not(not(x)), not(x)))), not(x))), implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x))), implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x))), implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.40  = { by axiom 1 (ifeq_axiom) }
% 30.81/4.40    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(implies(not(not(x)), not(x)))), not(x))), implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x))), implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))))), true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x))), implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.41  = { by axiom 5 (cn_1) }
% 30.81/4.41    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(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x))), implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.41  = { by axiom 1 (ifeq_axiom) }
% 30.81/4.41    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x))), implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), not(x)), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true, is_a_theorem(implies(not(implies(not(not(x)), not(x))), implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.41  = { by axiom 4 (condensed_detachment) }
% 30.81/4.42    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(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.42  = { by axiom 1 (ifeq_axiom) }
% 30.81/4.42    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.42  = { by axiom 1 (ifeq_axiom) R->L }
% 30.81/4.43    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(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.43  = { by axiom 4 (condensed_detachment) R->L }
% 30.81/4.44    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x)))), implies(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))))), true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.44  = { by axiom 5 (cn_1) }
% 30.81/4.44    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.44  = { by axiom 1 (ifeq_axiom) }
% 30.81/4.44    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.45  = { by axiom 5 (cn_1) }
% 30.81/4.45    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(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.45  = { by axiom 1 (ifeq_axiom) }
% 30.81/4.45    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x))))), not(x)), implies(not(not(x)), not(x))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(not(x)), implies(not(not(not(implies(not(not(x)), not(x))))), not(not(implies(not(not(x)), not(x)))))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.45  = { by axiom 4 (condensed_detachment) }
% 30.81/4.45    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.45  = { by axiom 1 (ifeq_axiom) }
% 30.81/4.45    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.46  = { by axiom 1 (ifeq_axiom) R->L }
% 30.81/4.46    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.46  = { by axiom 5 (cn_1) R->L }
% 30.81/4.46    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x)))), implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true, ifeq(is_a_theorem(implies(not(not(implies(not(not(x)), not(x)))), implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.46  = { by axiom 4 (condensed_detachment) }
% 30.81/4.46    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.46  = { by axiom 1 (ifeq_axiom) }
% 30.81/4.46    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.47  = { by axiom 1 (ifeq_axiom) R->L }
% 30.81/4.47    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.47  = { by axiom 4 (condensed_detachment) R->L }
% 30.81/4.47    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.48  = { by axiom 1 (ifeq_axiom) R->L }
% 30.81/4.48    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.48  = { by axiom 5 (cn_1) R->L }
% 30.81/4.49    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.49  = { by axiom 1 (ifeq_axiom) R->L }
% 30.81/4.49    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.49  = { by axiom 5 (cn_1) R->L }
% 30.81/4.50    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), implies(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.50  = { by axiom 4 (condensed_detachment) }
% 30.81/4.50    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.50  = { by axiom 1 (ifeq_axiom) }
% 30.81/4.50    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.51  = { by axiom 1 (ifeq_axiom) R->L }
% 30.81/4.51    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.51  = { by axiom 4 (condensed_detachment) R->L }
% 30.81/4.51    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.51  = { by axiom 3 (cn_2) }
% 30.81/4.52    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true, ifeq(true, true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 30.81/4.52  = { by axiom 1 (ifeq_axiom) }
% 32.44/4.52    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.52  = { by axiom 5 (cn_1) }
% 32.44/4.52    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.53  = { by axiom 1 (ifeq_axiom) }
% 32.44/4.53    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.53  = { by axiom 1 (ifeq_axiom) R->L }
% 32.44/4.53    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.54  = { by axiom 5 (cn_1) R->L }
% 32.44/4.54    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))))), true, ifeq(is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.54  = { by axiom 4 (condensed_detachment) }
% 32.44/4.54    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.54  = { by axiom 1 (ifeq_axiom) }
% 32.44/4.54    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), implies(not(not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(not(implies(not(not(x)), not(x))), not(implies(not(not(x)), not(x)))), implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.54  = { by axiom 4 (condensed_detachment) }
% 32.44/4.54    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.54  = { by axiom 1 (ifeq_axiom) }
% 32.44/4.54    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.54  = { by axiom 1 (ifeq_axiom) R->L }
% 32.44/4.54    ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.54  = { by axiom 4 (condensed_detachment) R->L }
% 32.44/4.54    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x))))), implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x))))))), true, ifeq(is_a_theorem(implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.54  = { by axiom 5 (cn_1) }
% 32.44/4.54    ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.54  = { by axiom 1 (ifeq_axiom) }
% 32.44/4.54    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.54  = { by axiom 1 (ifeq_axiom) R->L }
% 32.44/4.54    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.55  = { by axiom 4 (condensed_detachment) R->L }
% 32.44/4.55    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x))))))), true, ifeq(is_a_theorem(implies(implies(not(not(x)), not(x)), not(x))), true, is_a_theorem(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.55  = { by axiom 3 (cn_2) }
% 32.44/4.55    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x))))))), true, ifeq(true, true, is_a_theorem(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.55  = { by axiom 1 (ifeq_axiom) }
% 32.44/4.55    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(x)), implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x))))))), true, is_a_theorem(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.55  = { by axiom 5 (cn_1) }
% 32.44/4.55    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.56  = { by axiom 1 (ifeq_axiom) }
% 32.44/4.56    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.56  = { by axiom 1 (ifeq_axiom) R->L }
% 32.44/4.56    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.56  = { by axiom 4 (condensed_detachment) R->L }
% 32.44/4.56    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(x, implies(not(x), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x))))), implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))))), true, ifeq(is_a_theorem(implies(x, implies(not(x), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x))))), implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.56  = { by axiom 2 (cn_3) }
% 32.44/4.56    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(x, implies(not(x), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x))))), implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x))))), implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.57  = { by axiom 1 (ifeq_axiom) }
% 32.44/4.57    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(x, implies(not(x), not(implies(not(not(x)), not(x))))), implies(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x))))), implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))))), true, is_a_theorem(implies(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x))))), implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.57  = { by axiom 5 (cn_1) }
% 32.44/4.57    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x))))), implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x))))))), true), true, ifeq(is_a_theorem(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.57  = { by axiom 1 (ifeq_axiom) }
% 32.44/4.57    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x))))), implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x))))))), true, ifeq(is_a_theorem(implies(implies(not(x), not(implies(not(not(x)), not(x)))), implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true, is_a_theorem(implies(x, implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.57  = { by axiom 4 (condensed_detachment) }
% 32.44/4.57    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.57  = { by axiom 1 (ifeq_axiom) }
% 32.44/4.57    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x)))), implies(x, not(implies(not(not(x)), not(x)))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), not(implies(not(not(x)), not(x)))), not(implies(not(not(x)), not(x))))), true, is_a_theorem(implies(x, not(implies(not(not(x)), not(x))))), true), true), true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.57  = { by axiom 4 (condensed_detachment) }
% 32.44/4.57    ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.57  = { by axiom 1 (ifeq_axiom) }
% 32.44/4.57    ifeq(is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true, is_a_theorem(implies(not(x), implies(x, y))), true)
% 32.44/4.57  = { by axiom 1 (ifeq_axiom) R->L }
% 32.44/4.57    ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true, is_a_theorem(implies(not(x), implies(x, y))), true), true)
% 32.44/4.57  = { by axiom 4 (condensed_detachment) R->L }
% 32.44/4.57    ifeq(ifeq(is_a_theorem(implies(implies(not(x), implies(not(not(x)), not(x))), implies(implies(implies(not(not(x)), not(x)), implies(x, y)), implies(not(x), implies(x, y))))), true, ifeq(is_a_theorem(implies(not(x), implies(not(not(x)), not(x)))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), implies(x, y)), implies(not(x), implies(x, y)))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true, is_a_theorem(implies(not(x), implies(x, y))), true), true)
% 32.44/4.57  = { by axiom 2 (cn_3) }
% 32.44/4.57    ifeq(ifeq(is_a_theorem(implies(implies(not(x), implies(not(not(x)), not(x))), implies(implies(implies(not(not(x)), not(x)), implies(x, y)), implies(not(x), implies(x, y))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), implies(x, y)), implies(not(x), implies(x, y)))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true, is_a_theorem(implies(not(x), implies(x, y))), true), true)
% 32.44/4.57  = { by axiom 1 (ifeq_axiom) }
% 32.44/4.57    ifeq(ifeq(is_a_theorem(implies(implies(not(x), implies(not(not(x)), not(x))), implies(implies(implies(not(not(x)), not(x)), implies(x, y)), implies(not(x), implies(x, y))))), true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), implies(x, y)), implies(not(x), implies(x, y)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true, is_a_theorem(implies(not(x), implies(x, y))), true), true)
% 32.44/4.57  = { by axiom 5 (cn_1) }
% 32.44/4.57    ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(x)), not(x)), implies(x, y)), implies(not(x), implies(x, y)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true, is_a_theorem(implies(not(x), implies(x, y))), true), true)
% 32.44/4.57  = { by axiom 1 (ifeq_axiom) }
% 32.44/4.57    ifeq(is_a_theorem(implies(implies(implies(not(not(x)), not(x)), implies(x, y)), implies(not(x), implies(x, y)))), true, ifeq(is_a_theorem(implies(implies(not(not(x)), not(x)), implies(x, y))), true, is_a_theorem(implies(not(x), implies(x, y))), true), true)
% 32.44/4.57  = { by axiom 4 (condensed_detachment) }
% 32.44/4.57    true
% 32.44/4.57  % SZS output end Proof
% 32.44/4.57  
% 32.44/4.57  RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------