Twee---2.7.UNS-Prf.s

%------------------------------------------------------------------------------
% File     : Twee---2.7
% Problem  : LCL055-1 : TPTP v9.3.1. Released v1.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n016.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8046.5625MB
% OS       : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Tue Sep 29 11:48:55 AM UTC 2026

% Result   : Unsatisfiable 27.30s 3.96s
% Output   : Proof 40.64s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03  % Problem  : LCL055-1 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.04  % Command  : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.09/0.36  % Computer : n016.cluster.edu
% 0.09/0.36  % Model    : x86_64 x86_64
% 0.09/0.36  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.36  % Memory   : 8046.5625MB
% 0.09/0.36  % OS       : Linux 6.8.0-71-generic
% 0.09/0.36  % CPULimit : 300
% 0.09/0.36  % WCLimit  : 300
% 0.09/0.36  % DateTime : Sun Sep 27 15:24:45 UTC 2026
% 0.13/0.36  % CPUTime  : 
% 0.13/0.36  Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 27.30/3.96  Command-line arguments: --no-flatten-goal
% 27.30/3.96  
% 27.30/3.96  % SZS status Unsatisfiable
% 27.30/3.96  
% 36.19/5.07  % SZS output start Proof
% 36.19/5.07  Axiom 1 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 36.19/5.07  Axiom 2 (cn_3): is_a_theorem(implies(X, implies(not(X), Y))) = true.
% 36.19/5.07  Axiom 3 (cn_2): is_a_theorem(implies(implies(not(X), X), X)) = true.
% 36.19/5.07  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.
% 36.19/5.10  Axiom 5 (cn_1): is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
% 36.19/5.10  
% 36.19/5.10  Goal 1 (prove_cn_37): is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))) = true.
% 36.19/5.10  Proof:
% 36.19/5.10    is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c)))
% 36.19/5.10  = { by axiom 1 (ifeq_axiom) R->L }
% 36.19/5.10    ifeq(true, true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.10  = { by axiom 4 (condensed_detachment) R->L }
% 36.19/5.10    ifeq(ifeq(is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.10  = { by axiom 1 (ifeq_axiom) R->L }
% 36.19/5.10    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.10  = { by axiom 4 (condensed_detachment) R->L }
% 36.19/5.10    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b))), implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))))), true, ifeq(is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true), true), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.10  = { by axiom 5 (cn_1) }
% 36.19/5.10    ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true), true), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.10  = { by axiom 1 (ifeq_axiom) }
% 36.19/5.10    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.10  = { by axiom 5 (cn_1) }
% 36.19/5.10    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.11  = { by axiom 1 (ifeq_axiom) }
% 36.19/5.11    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.11  = { by axiom 1 (ifeq_axiom) R->L }
% 36.19/5.11    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, ifeq(true, true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.11  = { by axiom 4 (condensed_detachment) R->L }
% 36.19/5.11    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(not(implies(not(not(a)), not(a))), b)), implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(not(implies(not(not(a)), not(a))), b))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.11  = { by axiom 2 (cn_3) }
% 36.19/5.11    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(not(implies(not(not(a)), not(a))), b)), implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.12  = { by axiom 1 (ifeq_axiom) }
% 36.19/5.12    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(not(implies(not(not(a)), not(a))), b)), implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.12  = { by axiom 5 (cn_1) }
% 36.19/5.12    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.12  = { by axiom 1 (ifeq_axiom) }
% 36.19/5.12    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b))), implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b))))), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), b), implies(a, b)), implies(implies(not(not(a)), not(a)), implies(a, b)))), true, is_a_theorem(implies(implies(a, not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), implies(a, b)))), true), true), true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.12  = { by axiom 4 (condensed_detachment) }
% 36.19/5.12    ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.12  = { by axiom 1 (ifeq_axiom) }
% 36.19/5.12    ifeq(ifeq(is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.12  = { by axiom 1 (ifeq_axiom) R->L }
% 36.19/5.12    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.12  = { by axiom 4 (condensed_detachment) R->L }
% 36.19/5.12    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.12  = { by axiom 1 (ifeq_axiom) R->L }
% 36.19/5.13    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.13  = { by axiom 4 (condensed_detachment) R->L }
% 36.19/5.13    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.13  = { by axiom 2 (cn_3) }
% 36.19/5.13    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.13  = { by axiom 1 (ifeq_axiom) }
% 36.19/5.13    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.19/5.13  = { by axiom 5 (cn_1) }
% 36.19/5.14    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.14  = { by axiom 1 (ifeq_axiom) }
% 36.99/5.14    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.14  = { by axiom 1 (ifeq_axiom) R->L }
% 36.99/5.14    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true, ifeq(true, true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.14  = { by axiom 3 (cn_2) R->L }
% 36.99/5.14    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.14  = { by axiom 4 (condensed_detachment) }
% 36.99/5.14    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.15  = { by axiom 1 (ifeq_axiom) }
% 36.99/5.15    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.15  = { by axiom 1 (ifeq_axiom) R->L }
% 36.99/5.15    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.16  = { by axiom 4 (condensed_detachment) R->L }
% 36.99/5.16    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.16  = { by axiom 1 (ifeq_axiom) R->L }
% 36.99/5.16    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.16  = { by axiom 4 (condensed_detachment) R->L }
% 36.99/5.17    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.17  = { by axiom 3 (cn_2) }
% 36.99/5.17    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.17  = { by axiom 1 (ifeq_axiom) }
% 36.99/5.17    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.18  = { by axiom 5 (cn_1) }
% 36.99/5.18    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.18  = { by axiom 1 (ifeq_axiom) }
% 36.99/5.18    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.18  = { by axiom 1 (ifeq_axiom) R->L }
% 36.99/5.18    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.18  = { by axiom 5 (cn_1) R->L }
% 36.99/5.19    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a))))), implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.19  = { by axiom 4 (condensed_detachment) }
% 36.99/5.19    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.19  = { by axiom 1 (ifeq_axiom) }
% 36.99/5.19    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.19  = { by axiom 1 (ifeq_axiom) R->L }
% 36.99/5.19    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 36.99/5.20  = { by axiom 4 (condensed_detachment) R->L }
% 36.99/5.20    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 37.49/5.21  = { by axiom 1 (ifeq_axiom) R->L }
% 37.49/5.21    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 37.49/5.21  = { by axiom 4 (condensed_detachment) R->L }
% 37.49/5.22    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 37.49/5.22  = { by axiom 1 (ifeq_axiom) R->L }
% 37.49/5.23    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 37.49/5.23  = { by axiom 5 (cn_1) R->L }
% 37.49/5.24    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(a))), not(not(implies(not(not(a)), not(a)))))), implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, is_a_theorem(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 37.49/5.24  = { by axiom 1 (ifeq_axiom) R->L }
% 37.49/5.24    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(a))), not(not(implies(not(not(a)), not(a)))))), implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 37.49/5.25  = { by axiom 2 (cn_3) R->L }
% 37.49/5.25    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(a))), not(not(implies(not(not(a)), not(a)))))), implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(is_a_theorem(implies(not(not(a)), implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 37.49/5.25  = { by axiom 4 (condensed_detachment) }
% 37.49/5.26    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(true, true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 37.49/5.26  = { by axiom 1 (ifeq_axiom) }
% 37.49/5.26    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 37.49/5.27  = { by axiom 1 (ifeq_axiom) R->L }
% 37.49/5.27    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 37.49/5.28  = { by axiom 4 (condensed_detachment) R->L }
% 37.49/5.29    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 37.49/5.29  = { by axiom 1 (ifeq_axiom) R->L }
% 37.49/5.30    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 38.26/5.31  = { by axiom 5 (cn_1) R->L }
% 38.26/5.31    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a))))), implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))))), true, is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 38.26/5.31  = { by axiom 1 (ifeq_axiom) R->L }
% 38.26/5.32    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a))))), implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 38.26/5.32  = { by axiom 2 (cn_3) R->L }
% 38.26/5.33    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a))))), implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))))), true, ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 38.26/5.34  = { by axiom 4 (condensed_detachment) }
% 38.26/5.34    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 38.26/5.34  = { by axiom 1 (ifeq_axiom) }
% 38.26/5.35    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 38.26/5.35  = { by axiom 5 (cn_1) }
% 38.26/5.35    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 38.26/5.36  = { by axiom 1 (ifeq_axiom) }
% 38.26/5.36    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 38.26/5.36  = { by axiom 1 (ifeq_axiom) R->L }
% 38.26/5.37    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 38.26/5.37  = { by axiom 5 (cn_1) R->L }
% 38.26/5.37    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))))), true, ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(implies(not(not(a)), not(a))))), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 38.26/5.37  = { by axiom 4 (condensed_detachment) }
% 38.26/5.38    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 38.26/5.38  = { by axiom 1 (ifeq_axiom) }
% 38.26/5.38    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 38.26/5.38  = { by axiom 1 (ifeq_axiom) R->L }
% 38.26/5.38    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 38.26/5.39  = { by axiom 5 (cn_1) R->L }
% 38.26/5.39    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))))), implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 38.26/5.39  = { by axiom 4 (condensed_detachment) }
% 38.26/5.39    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 38.26/5.39  = { by axiom 1 (ifeq_axiom) }
% 38.26/5.39    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 38.26/5.40  = { by axiom 1 (ifeq_axiom) R->L }
% 38.26/5.40    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.40  = { by axiom 4 (condensed_detachment) R->L }
% 39.06/5.40    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true, ifeq(is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.41  = { by axiom 5 (cn_1) }
% 39.06/5.41    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.41  = { by axiom 1 (ifeq_axiom) }
% 39.06/5.41    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.42  = { by axiom 1 (ifeq_axiom) R->L }
% 39.06/5.42    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.42  = { by axiom 4 (condensed_detachment) R->L }
% 39.06/5.42    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))))), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.43  = { by axiom 3 (cn_2) }
% 39.06/5.43    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))))), true, ifeq(true, true, is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.43  = { by axiom 1 (ifeq_axiom) }
% 39.06/5.43    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))))), true, is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.44  = { by axiom 5 (cn_1) }
% 39.06/5.44    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.45  = { by axiom 1 (ifeq_axiom) }
% 39.06/5.45    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.45  = { by axiom 1 (ifeq_axiom) R->L }
% 39.06/5.45    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.46  = { by axiom 4 (condensed_detachment) R->L }
% 39.06/5.46    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(implies(not(not(a)), not(a)))), not(a))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))))), true, ifeq(is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(not(not(implies(not(not(a)), not(a)))), not(a)))), true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.46  = { by axiom 2 (cn_3) }
% 39.06/5.47    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(implies(not(not(a)), not(a)))), not(a))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.47  = { by axiom 1 (ifeq_axiom) }
% 39.06/5.47    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(implies(not(not(a)), not(a)))), not(a))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))))), true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.47  = { by axiom 5 (cn_1) }
% 39.06/5.48    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.48  = { by axiom 1 (ifeq_axiom) }
% 39.06/5.48    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))), implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(a)), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true, is_a_theorem(implies(not(implies(not(not(a)), not(a))), implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.49  = { by axiom 4 (condensed_detachment) }
% 39.06/5.49    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.49  = { by axiom 1 (ifeq_axiom) }
% 39.06/5.49    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.06/5.50  = { by axiom 1 (ifeq_axiom) R->L }
% 39.85/5.50    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.50  = { by axiom 4 (condensed_detachment) R->L }
% 39.85/5.51    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a)))), implies(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.51  = { by axiom 5 (cn_1) }
% 39.85/5.51    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.51  = { by axiom 1 (ifeq_axiom) }
% 39.85/5.52    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.52  = { by axiom 5 (cn_1) }
% 39.85/5.52    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.52  = { by axiom 1 (ifeq_axiom) }
% 39.85/5.52    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a))))), not(a)), implies(not(not(a)), not(a))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(not(a)), implies(not(not(not(implies(not(not(a)), not(a))))), not(not(implies(not(not(a)), not(a)))))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.53  = { by axiom 4 (condensed_detachment) }
% 39.85/5.53    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.53  = { by axiom 1 (ifeq_axiom) }
% 39.85/5.53    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.53  = { by axiom 1 (ifeq_axiom) R->L }
% 39.85/5.53    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.53  = { by axiom 5 (cn_1) R->L }
% 39.85/5.53    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a)))), implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(not(not(implies(not(not(a)), not(a)))), implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.53  = { by axiom 4 (condensed_detachment) }
% 39.85/5.53    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.54  = { by axiom 1 (ifeq_axiom) }
% 39.85/5.54    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.54  = { by axiom 1 (ifeq_axiom) R->L }
% 39.85/5.54    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.55  = { by axiom 4 (condensed_detachment) R->L }
% 39.85/5.55    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.55  = { by axiom 1 (ifeq_axiom) R->L }
% 39.85/5.55    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.56  = { by axiom 5 (cn_1) R->L }
% 39.85/5.56    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.56  = { by axiom 1 (ifeq_axiom) R->L }
% 39.85/5.57    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.57  = { by axiom 5 (cn_1) R->L }
% 39.85/5.57    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), implies(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.58  = { by axiom 4 (condensed_detachment) }
% 39.85/5.58    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.58  = { by axiom 1 (ifeq_axiom) }
% 39.85/5.58    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.59  = { by axiom 1 (ifeq_axiom) R->L }
% 39.85/5.59    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.59  = { by axiom 4 (condensed_detachment) R->L }
% 39.85/5.59    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 39.85/5.60  = { by axiom 3 (cn_2) }
% 40.64/5.60    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, ifeq(true, true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.60  = { by axiom 1 (ifeq_axiom) }
% 40.64/5.61    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.61  = { by axiom 5 (cn_1) }
% 40.64/5.61    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.62  = { by axiom 1 (ifeq_axiom) }
% 40.64/5.62    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.62  = { by axiom 1 (ifeq_axiom) R->L }
% 40.64/5.62    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.62  = { by axiom 5 (cn_1) R->L }
% 40.64/5.62    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))))), true, ifeq(is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.62  = { by axiom 4 (condensed_detachment) }
% 40.64/5.62    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.62  = { by axiom 1 (ifeq_axiom) }
% 40.64/5.62    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), implies(not(not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(not(implies(not(not(a)), not(a))), not(implies(not(not(a)), not(a)))), implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.62  = { by axiom 4 (condensed_detachment) }
% 40.64/5.62    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.62  = { by axiom 1 (ifeq_axiom) }
% 40.64/5.62    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.62  = { by axiom 1 (ifeq_axiom) R->L }
% 40.64/5.63    ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.63  = { by axiom 4 (condensed_detachment) R->L }
% 40.64/5.63    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.63  = { by axiom 5 (cn_1) }
% 40.64/5.63    ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.63  = { by axiom 1 (ifeq_axiom) }
% 40.64/5.63    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.63  = { by axiom 1 (ifeq_axiom) R->L }
% 40.64/5.63    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.63  = { by axiom 4 (condensed_detachment) R->L }
% 40.64/5.63    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(a)), implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), not(a))), true, is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.63  = { by axiom 3 (cn_2) }
% 40.64/5.63    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(a)), implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))))), true, ifeq(true, true, is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.63  = { by axiom 1 (ifeq_axiom) }
% 40.64/5.64    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(a)), implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))))), true, is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.64  = { by axiom 5 (cn_1) }
% 40.64/5.64    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.64  = { by axiom 1 (ifeq_axiom) }
% 40.64/5.64    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.64  = { by axiom 1 (ifeq_axiom) R->L }
% 40.64/5.64    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.64  = { by axiom 4 (condensed_detachment) R->L }
% 40.64/5.64    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, implies(not(a), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))))), true, ifeq(is_a_theorem(implies(a, implies(not(a), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.64  = { by axiom 2 (cn_3) }
% 40.64/5.65    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, implies(not(a), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.65  = { by axiom 1 (ifeq_axiom) }
% 40.64/5.65    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, implies(not(a), not(implies(not(not(a)), not(a))))), implies(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))))), true, is_a_theorem(implies(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.65  = { by axiom 5 (cn_1) }
% 40.64/5.65    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))))), true), true, ifeq(is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.65  = { by axiom 1 (ifeq_axiom) }
% 40.64/5.65    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))), implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a))))))), true, ifeq(is_a_theorem(implies(implies(not(a), not(implies(not(not(a)), not(a)))), implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true, is_a_theorem(implies(a, implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.65  = { by axiom 4 (condensed_detachment) }
% 40.64/5.65    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.65  = { by axiom 1 (ifeq_axiom) }
% 40.64/5.65    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a)))), implies(a, not(implies(not(not(a)), not(a)))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), not(implies(not(not(a)), not(a)))), not(implies(not(not(a)), not(a))))), true, is_a_theorem(implies(a, not(implies(not(not(a)), not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.65  = { by axiom 4 (condensed_detachment) }
% 40.64/5.65    ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.65  = { by axiom 1 (ifeq_axiom) }
% 40.64/5.65    ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true)
% 40.64/5.65  = { by axiom 1 (ifeq_axiom) R->L }
% 40.64/5.65    ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true), true)
% 40.64/5.65  = { by axiom 4 (condensed_detachment) R->L }
% 40.64/5.65    ifeq(ifeq(is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))))), true, ifeq(is_a_theorem(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c)))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c)))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true), true)
% 40.64/5.65  = { by axiom 5 (cn_1) }
% 40.64/5.65    ifeq(ifeq(is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c)))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true), true)
% 40.64/5.65  = { by axiom 1 (ifeq_axiom) }
% 40.64/5.66    ifeq(ifeq(is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true), true)
% 40.64/5.66  = { by axiom 1 (ifeq_axiom) R->L }
% 40.64/5.66    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true), true)
% 40.64/5.66  = { by axiom 4 (condensed_detachment) R->L }
% 40.64/5.66    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(a), implies(not(not(a)), not(a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b))))), true, ifeq(is_a_theorem(implies(not(a), implies(not(not(a)), not(a)))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b)))), true), true), true, is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true), true)
% 40.64/5.66  = { by axiom 2 (cn_3) }
% 40.64/5.66    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(a), implies(not(not(a)), not(a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b)))), true), true), true, is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true), true)
% 40.64/5.66  = { by axiom 1 (ifeq_axiom) }
% 40.64/5.66    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(a), implies(not(not(a)), not(a))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b))))), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b)))), true), true, is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true), true)
% 40.64/5.66  = { by axiom 5 (cn_1) }
% 40.64/5.66    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b)))), true), true, is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true), true)
% 40.64/5.66  = { by axiom 1 (ifeq_axiom) }
% 40.64/5.66    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b)))), true, is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))))), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true), true)
% 40.64/5.66  = { by axiom 1 (ifeq_axiom) R->L }
% 40.64/5.66    ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b)))), true, is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true), true)
% 40.64/5.66  = { by axiom 5 (cn_1) R->L }
% 40.64/5.66    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b))), implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c)))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(not(a), implies(a, b)))), true, is_a_theorem(implies(implies(implies(not(a), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))), implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true), true)
% 40.64/5.66  = { by axiom 4 (condensed_detachment) }
% 40.64/5.66    ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true), true)
% 40.64/5.66  = { by axiom 1 (ifeq_axiom) }
% 40.64/5.66    ifeq(is_a_theorem(implies(implies(implies(not(not(a)), not(a)), implies(a, b)), implies(implies(implies(a, b), c), implies(not(a), c)))), true, ifeq(is_a_theorem(implies(implies(not(not(a)), not(a)), implies(a, b))), true, is_a_theorem(implies(implies(implies(a, b), c), implies(not(a), c))), true), true)
% 40.64/5.66  = { by axiom 4 (condensed_detachment) }
% 40.64/5.66    true
% 40.64/5.66  % SZS output end Proof
% 40.64/5.66  
% 40.64/5.66  RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------