Twee---2.7.UNS-Prf.s

%------------------------------------------------------------------------------
% File     : Twee---2.7
% Problem  : LCL366-1 : TPTP v9.3.1. Bugfixed v5.5.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p

% Computer : n013.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8046.5625MB
% OS       : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Tue Sep 29 11:49:40 AM UTC 2026

% Result   : Unsatisfiable 26.52s 3.84s
% Output   : Proof 39.13s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03  % Problem  : LCL366-1 : TPTP v9.3.1. Bugfixed v5.5.0.
% 0.00/0.04  % Command  : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.38  % Computer : n013.cluster.edu
% 0.09/0.38  % Model    : x86_64 x86_64
% 0.09/0.38  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.38  % Memory   : 8046.5625MB
% 0.09/0.38  % OS       : Linux 6.8.0-71-generic
% 0.09/0.38  % CPULimit : 300
% 0.09/0.38  % WCLimit  : 300
% 0.09/0.38  % DateTime : Sun Sep 27 15:36:21 UTC 2026
% 0.09/0.39  % CPUTime  : 
% 0.09/0.39  Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 26.52/3.84  Command-line arguments: --no-flatten-goal
% 26.52/3.84  
% 26.52/3.84  % SZS status Unsatisfiable
% 26.52/3.84  
% 33.67/4.77  % SZS output start Proof
% 33.67/4.77  Axiom 1 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 33.67/4.77  Axiom 2 (cn_3): is_a_theorem(implies(X, implies(not(X), Y))) = true.
% 33.67/4.77  Axiom 3 (cn_2): is_a_theorem(implies(implies(not(X), X), X)) = true.
% 33.67/4.77  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.
% 33.67/4.80  Axiom 5 (cn_1): is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
% 33.67/4.80  
% 33.67/4.80  Goal 1 (prove_cn_15): is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))) = true.
% 33.67/4.80  Proof:
% 33.67/4.80    is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x)))
% 33.67/4.80  = { by axiom 1 (ifeq_axiom) R->L }
% 33.67/4.80    ifeq(true, true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 33.67/4.80  = { by axiom 4 (condensed_detachment) R->L }
% 33.67/4.80    ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))))), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 33.67/4.80  = { by axiom 1 (ifeq_axiom) R->L }
% 33.67/4.80    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 33.67/4.80  = { by axiom 5 (cn_1) R->L }
% 33.67/4.80    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), implies(not(x), x)))), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 33.67/4.80  = { by axiom 1 (ifeq_axiom) R->L }
% 33.67/4.80    ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), implies(not(x), x)))), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 33.67/4.80  = { by axiom 5 (cn_1) R->L }
% 34.47/4.80    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), implies(not(x), x))), implies(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), implies(not(x), x)))), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.80  = { by axiom 4 (condensed_detachment) }
% 34.47/4.80    ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.80  = { by axiom 1 (ifeq_axiom) }
% 34.47/4.80    ifeq(ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.80  = { by axiom 1 (ifeq_axiom) R->L }
% 34.47/4.80    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.80  = { by axiom 4 (condensed_detachment) R->L }
% 34.47/4.81    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), x), x)), implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))))), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.81  = { by axiom 5 (cn_1) }
% 34.47/4.81    ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.81  = { by axiom 1 (ifeq_axiom) }
% 34.47/4.81    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.81  = { by axiom 1 (ifeq_axiom) R->L }
% 34.47/4.81    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.81  = { by axiom 4 (condensed_detachment) R->L }
% 34.47/4.81    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), x), x), implies(implies(x, x), implies(implies(not(x), x), x)))), true, ifeq(is_a_theorem(implies(implies(not(x), x), x)), true, is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.81  = { by axiom 3 (cn_2) }
% 34.47/4.81    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), x), x), implies(implies(x, x), implies(implies(not(x), x), x)))), true, ifeq(true, true, is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.81  = { by axiom 1 (ifeq_axiom) }
% 34.47/4.81    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), x), x), implies(implies(x, x), implies(implies(not(x), x), x)))), true, is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.81  = { by axiom 5 (cn_1) }
% 34.47/4.81    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.81  = { by axiom 1 (ifeq_axiom) }
% 34.47/4.82    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.82  = { by axiom 1 (ifeq_axiom) R->L }
% 34.47/4.82    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.82  = { by axiom 4 (condensed_detachment) R->L }
% 34.47/4.82    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), y), implies(x, x)), implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x))))), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(x, x))), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.82  = { by axiom 5 (cn_1) }
% 34.47/4.82    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(x, x))), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.82  = { by axiom 1 (ifeq_axiom) }
% 34.47/4.82    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(x), y), implies(x, x))), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.82  = { by axiom 1 (ifeq_axiom) R->L }
% 34.47/4.82    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.82  = { by axiom 4 (condensed_detachment) R->L }
% 34.47/4.82    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x))), implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x))))), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.83  = { by axiom 5 (cn_1) }
% 34.47/4.83    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.83  = { by axiom 1 (ifeq_axiom) }
% 34.47/4.83    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.83  = { by axiom 1 (ifeq_axiom) R->L }
% 34.47/4.83    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.83  = { by axiom 4 (condensed_detachment) R->L }
% 34.47/4.83    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(x, x)), not(implies(not(x), y))), implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x))))), true, ifeq(is_a_theorem(implies(not(implies(x, x)), not(implies(not(x), y)))), true, is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.83  = { by axiom 5 (cn_1) }
% 34.47/4.84    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(implies(x, x)), not(implies(not(x), y)))), true, is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.84  = { by axiom 1 (ifeq_axiom) }
% 34.47/4.84    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(implies(x, x)), not(implies(not(x), y)))), true, is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.84  = { by axiom 1 (ifeq_axiom) R->L }
% 34.47/4.84    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(x, x)), not(implies(not(x), y)))), true), true, is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.84  = { by axiom 4 (condensed_detachment) R->L }
% 34.47/4.84    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), x), x), implies(x, x))), true, ifeq(is_a_theorem(implies(implies(not(x), x), x)), true, is_a_theorem(implies(x, x)), true), true), true, is_a_theorem(implies(not(implies(x, x)), not(implies(not(x), y)))), true), true, is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.85  = { by axiom 3 (cn_2) }
% 34.47/4.85    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), x), x), implies(x, x))), true, ifeq(true, true, is_a_theorem(implies(x, x)), true), true), true, is_a_theorem(implies(not(implies(x, x)), not(implies(not(x), y)))), true), true, is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.85  = { by axiom 1 (ifeq_axiom) }
% 34.47/4.85    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), x), x), implies(x, x))), true, is_a_theorem(implies(x, x)), true), true, is_a_theorem(implies(not(implies(x, x)), not(implies(not(x), y)))), true), true, is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.85  = { by axiom 1 (ifeq_axiom) R->L }
% 34.47/4.85    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(x), x), x), implies(x, x))), true), true, is_a_theorem(implies(x, x)), true), true, is_a_theorem(implies(not(implies(x, x)), not(implies(not(x), y)))), true), true, is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.85  = { by axiom 5 (cn_1) R->L }
% 34.47/4.86    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(x, implies(not(x), x)), implies(implies(implies(not(x), x), x), implies(x, x)))), true, is_a_theorem(implies(implies(implies(not(x), x), x), implies(x, x))), true), true, is_a_theorem(implies(x, x)), true), true, is_a_theorem(implies(not(implies(x, x)), not(implies(not(x), y)))), true), true, is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.86  = { by axiom 1 (ifeq_axiom) R->L }
% 34.47/4.86    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(x, implies(not(x), x)), implies(implies(implies(not(x), x), x), implies(x, x)))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(x), x), x), implies(x, x))), true), true), true, is_a_theorem(implies(x, x)), true), true, is_a_theorem(implies(not(implies(x, x)), not(implies(not(x), y)))), true), true, is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.86  = { by axiom 2 (cn_3) R->L }
% 34.47/4.86    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(x, implies(not(x), x)), implies(implies(implies(not(x), x), x), implies(x, x)))), true, ifeq(is_a_theorem(implies(x, implies(not(x), x))), true, is_a_theorem(implies(implies(implies(not(x), x), x), implies(x, x))), true), true), true, is_a_theorem(implies(x, x)), true), true, is_a_theorem(implies(not(implies(x, x)), not(implies(not(x), y)))), true), true, is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.86  = { by axiom 4 (condensed_detachment) }
% 34.47/4.86    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(x, x)), true), true, is_a_theorem(implies(not(implies(x, x)), not(implies(not(x), y)))), true), true, is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.87  = { by axiom 1 (ifeq_axiom) }
% 34.47/4.87    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(x, x)), true, is_a_theorem(implies(not(implies(x, x)), not(implies(not(x), y)))), true), true, is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.87  = { by axiom 1 (ifeq_axiom) R->L }
% 34.47/4.87    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(x, x)), true, is_a_theorem(implies(not(implies(x, x)), not(implies(not(x), y)))), true), true), true, is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.87  = { by axiom 2 (cn_3) R->L }
% 34.47/4.87    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(x, x), implies(not(implies(x, x)), not(implies(not(x), y))))), true, ifeq(is_a_theorem(implies(x, x)), true, is_a_theorem(implies(not(implies(x, x)), not(implies(not(x), y)))), true), true), true, is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.87  = { by axiom 4 (condensed_detachment) }
% 34.47/4.88    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.88  = { by axiom 1 (ifeq_axiom) }
% 34.47/4.88    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.88  = { by axiom 1 (ifeq_axiom) R->L }
% 34.47/4.88    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.88  = { by axiom 4 (condensed_detachment) R->L }
% 34.47/4.88    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), y), implies(not(implies(not(x), y)), implies(x, x))), implies(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x))), implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))))), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(not(implies(not(x), y)), implies(x, x)))), true, is_a_theorem(implies(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x))), implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.88  = { by axiom 2 (cn_3) }
% 34.47/4.89    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), y), implies(not(implies(not(x), y)), implies(x, x))), implies(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x))), implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x))), implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.89  = { by axiom 1 (ifeq_axiom) }
% 34.47/4.89    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), y), implies(not(implies(not(x), y)), implies(x, x))), implies(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x))), implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))))), true, is_a_theorem(implies(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x))), implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x))))), true), true, ifeq(is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.89  = { by axiom 5 (cn_1) }
% 34.47/4.89    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x))), implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x))))), true), true, ifeq(is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.89  = { by axiom 1 (ifeq_axiom) }
% 34.47/4.89    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x))), implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x))))), true, ifeq(is_a_theorem(implies(implies(not(implies(not(x), y)), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true, is_a_theorem(implies(implies(not(x), y), implies(not(implies(x, x)), implies(x, x)))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.89  = { by axiom 4 (condensed_detachment) }
% 34.47/4.89    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.89  = { by axiom 1 (ifeq_axiom) }
% 34.47/4.89    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.89  = { by axiom 1 (ifeq_axiom) R->L }
% 34.47/4.90    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true, ifeq(true, true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.90  = { by axiom 3 (cn_2) R->L }
% 34.47/4.90    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(implies(not(x), y), implies(x, x)))), true, ifeq(is_a_theorem(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x))), true, is_a_theorem(implies(implies(not(x), y), implies(x, x))), true), true), true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.90  = { by axiom 4 (condensed_detachment) }
% 34.47/4.90    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.90  = { by axiom 1 (ifeq_axiom) }
% 34.47/4.90    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(x, x), implies(implies(not(x), x), x)), implies(implies(not(x), y), implies(implies(not(x), x), x)))), true, ifeq(is_a_theorem(implies(implies(x, x), implies(implies(not(x), x), x))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), x), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.90  = { by axiom 4 (condensed_detachment) }
% 34.47/4.90    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 34.47/4.90  = { by axiom 1 (ifeq_axiom) }
% 35.25/4.90    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 35.25/4.90  = { by axiom 1 (ifeq_axiom) R->L }
% 35.25/4.90    ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 35.25/4.90  = { by axiom 4 (condensed_detachment) R->L }
% 35.25/4.90    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(implies(not(x), x), x), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(implies(not(x), x), x), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 35.25/4.90  = { by axiom 5 (cn_1) }
% 35.25/4.90    ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(implies(not(x), x), x), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 35.25/4.90  = { by axiom 1 (ifeq_axiom) }
% 35.25/4.90    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(implies(not(x), x), x), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 35.25/4.90  = { by axiom 5 (cn_1) }
% 35.25/4.90    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 35.25/4.90  = { by axiom 1 (ifeq_axiom) }
% 35.25/4.90    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(x), x), x), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(y, x), implies(not(x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 35.25/4.90  = { by axiom 4 (condensed_detachment) }
% 35.25/4.90    ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 35.25/4.90  = { by axiom 1 (ifeq_axiom) }
% 35.25/4.90    ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true)
% 35.25/4.90  = { by axiom 1 (ifeq_axiom) R->L }
% 35.25/4.90    ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.91  = { by axiom 4 (condensed_detachment) R->L }
% 35.25/4.91    ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.91  = { by axiom 1 (ifeq_axiom) R->L }
% 35.25/4.91    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.91  = { by axiom 5 (cn_1) R->L }
% 35.25/4.91    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.91  = { by axiom 1 (ifeq_axiom) R->L }
% 35.25/4.91    ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.91  = { by axiom 5 (cn_1) R->L }
% 35.25/4.91    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, ifeq(is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.91  = { by axiom 4 (condensed_detachment) }
% 35.25/4.91    ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.91  = { by axiom 1 (ifeq_axiom) }
% 35.25/4.91    ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.92  = { by axiom 1 (ifeq_axiom) R->L }
% 35.25/4.92    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.92  = { by axiom 4 (condensed_detachment) R->L }
% 35.25/4.92    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.92  = { by axiom 1 (ifeq_axiom) R->L }
% 35.25/4.93    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.93  = { by axiom 4 (condensed_detachment) R->L }
% 35.25/4.93    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.94  = { by axiom 5 (cn_1) }
% 35.25/4.94    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.94  = { by axiom 1 (ifeq_axiom) }
% 35.25/4.94    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.95  = { by axiom 1 (ifeq_axiom) R->L }
% 35.25/4.95    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.95  = { by axiom 5 (cn_1) R->L }
% 35.25/4.96    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.96  = { by axiom 1 (ifeq_axiom) R->L }
% 35.25/4.96    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.96  = { by axiom 3 (cn_2) R->L }
% 35.25/4.97    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.97  = { by axiom 4 (condensed_detachment) }
% 35.25/4.97    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.98  = { by axiom 1 (ifeq_axiom) }
% 35.25/4.98    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.99  = { by axiom 1 (ifeq_axiom) R->L }
% 35.25/4.99    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/4.99  = { by axiom 4 (condensed_detachment) R->L }
% 35.25/5.00    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/5.01  = { by axiom 5 (cn_1) }
% 35.25/5.01    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/5.01  = { by axiom 1 (ifeq_axiom) }
% 35.25/5.02    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/5.02  = { by axiom 5 (cn_1) }
% 35.25/5.02    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/5.03  = { by axiom 1 (ifeq_axiom) }
% 35.25/5.03    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/5.03  = { by axiom 4 (condensed_detachment) }
% 35.25/5.03    ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/5.03  = { by axiom 1 (ifeq_axiom) }
% 35.25/5.03    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/5.04  = { by axiom 1 (ifeq_axiom) R->L }
% 35.25/5.04    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/5.04  = { by axiom 4 (condensed_detachment) R->L }
% 35.25/5.04    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 35.25/5.05  = { by axiom 1 (ifeq_axiom) R->L }
% 35.25/5.05    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 36.37/5.06  = { by axiom 4 (condensed_detachment) R->L }
% 36.37/5.06    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 36.37/5.07  = { by axiom 1 (ifeq_axiom) R->L }
% 36.37/5.08    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 36.37/5.08  = { by axiom 5 (cn_1) R->L }
% 36.37/5.08    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true, is_a_theorem(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 36.37/5.09  = { by axiom 1 (ifeq_axiom) R->L }
% 36.37/5.09    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 36.37/5.10  = { by axiom 2 (cn_3) R->L }
% 36.37/5.10    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true, ifeq(is_a_theorem(implies(implies(y, x), implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), true, is_a_theorem(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 36.37/5.11  = { by axiom 4 (condensed_detachment) }
% 36.37/5.11    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true, ifeq(true, true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 36.37/5.11  = { by axiom 1 (ifeq_axiom) }
% 36.37/5.12    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 36.37/5.12  = { by axiom 1 (ifeq_axiom) R->L }
% 36.37/5.12    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 36.37/5.13  = { by axiom 4 (condensed_detachment) R->L }
% 36.37/5.14    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 36.37/5.14  = { by axiom 1 (ifeq_axiom) R->L }
% 36.37/5.15    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 36.37/5.15  = { by axiom 5 (cn_1) R->L }
% 37.23/5.16    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x)))), implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))))), true, is_a_theorem(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.23/5.17  = { by axiom 1 (ifeq_axiom) R->L }
% 37.23/5.17    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x)))), implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.23/5.18  = { by axiom 2 (cn_3) R->L }
% 37.23/5.18    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x)))), implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))))), true, ifeq(is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x))))), true, is_a_theorem(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.23/5.19  = { by axiom 4 (condensed_detachment) }
% 37.23/5.19    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.23/5.20  = { by axiom 1 (ifeq_axiom) }
% 37.23/5.20    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(y, x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.20  = { by axiom 5 (cn_1) }
% 37.62/5.21    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.21  = { by axiom 1 (ifeq_axiom) }
% 37.62/5.21    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.21  = { by axiom 1 (ifeq_axiom) R->L }
% 37.62/5.22    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.22  = { by axiom 5 (cn_1) R->L }
% 37.62/5.22    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))))), true, ifeq(is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(y, x)), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))))), true), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.23  = { by axiom 4 (condensed_detachment) }
% 37.62/5.23    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.23  = { by axiom 1 (ifeq_axiom) }
% 37.62/5.23    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.24  = { by axiom 1 (ifeq_axiom) R->L }
% 37.62/5.24    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.24  = { by axiom 5 (cn_1) R->L }
% 37.62/5.24    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))))), implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))))), true, ifeq(is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))))), true, is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.25  = { by axiom 4 (condensed_detachment) }
% 37.62/5.25    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.25  = { by axiom 1 (ifeq_axiom) }
% 37.62/5.25    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.25  = { by axiom 1 (ifeq_axiom) R->L }
% 37.62/5.26    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.26  = { by axiom 4 (condensed_detachment) R->L }
% 37.62/5.26    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x)), implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true, ifeq(is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.27  = { by axiom 5 (cn_1) }
% 37.62/5.27    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.27  = { by axiom 1 (ifeq_axiom) }
% 37.62/5.27    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.28  = { by axiom 1 (ifeq_axiom) R->L }
% 37.62/5.28    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.28  = { by axiom 4 (condensed_detachment) R->L }
% 37.62/5.28    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x)))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.29  = { by axiom 3 (cn_2) }
% 37.62/5.29    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x)))), true, ifeq(true, true, is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 37.62/5.30  = { by axiom 1 (ifeq_axiom) }
% 37.62/5.30    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x)))), true, is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.30  = { by axiom 5 (cn_1) }
% 38.37/5.30    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.31  = { by axiom 1 (ifeq_axiom) }
% 38.37/5.31    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.31  = { by axiom 1 (ifeq_axiom) R->L }
% 38.37/5.32    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.32  = { by axiom 4 (condensed_detachment) R->L }
% 38.37/5.32    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x)), implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))))), true, ifeq(is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x))), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x)))), true), true), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.33  = { by axiom 2 (cn_3) }
% 38.37/5.33    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x)), implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x)))), true), true), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.33  = { by axiom 1 (ifeq_axiom) }
% 38.37/5.34    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x)), implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))))), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.34  = { by axiom 5 (cn_1) }
% 38.37/5.34    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.34  = { by axiom 1 (ifeq_axiom) }
% 38.37/5.35    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x)))), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), x), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.35  = { by axiom 4 (condensed_detachment) }
% 38.37/5.35    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.35  = { by axiom 1 (ifeq_axiom) }
% 38.37/5.36    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.36  = { by axiom 1 (ifeq_axiom) R->L }
% 38.37/5.36    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.37  = { by axiom 4 (condensed_detachment) R->L }
% 38.37/5.37    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))))), true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.37  = { by axiom 5 (cn_1) }
% 38.37/5.37    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.38  = { by axiom 1 (ifeq_axiom) }
% 38.37/5.38    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.38  = { by axiom 5 (cn_1) }
% 38.37/5.38    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.38  = { by axiom 1 (ifeq_axiom) }
% 38.37/5.39    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x)))), x), implies(implies(y, x), x)), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(y, x), implies(not(not(implies(implies(not(x), y), implies(implies(y, x), x)))), not(implies(implies(not(x), y), implies(implies(y, x), x))))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.39  = { by axiom 4 (condensed_detachment) }
% 38.37/5.39    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.39  = { by axiom 1 (ifeq_axiom) }
% 38.37/5.39    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.39  = { by axiom 1 (ifeq_axiom) R->L }
% 38.37/5.40    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.40  = { by axiom 5 (cn_1) R->L }
% 38.37/5.40    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, ifeq(is_a_theorem(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.40  = { by axiom 4 (condensed_detachment) }
% 38.37/5.40    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 38.37/5.40  = { by axiom 1 (ifeq_axiom) }
% 38.37/5.40    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 39.13/5.40  = { by axiom 1 (ifeq_axiom) R->L }
% 39.13/5.40    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 39.13/5.40  = { by axiom 4 (condensed_detachment) R->L }
% 39.13/5.41    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))))), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 39.13/5.41  = { by axiom 3 (cn_2) }
% 39.13/5.41    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 39.13/5.41  = { by axiom 1 (ifeq_axiom) }
% 39.13/5.41    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 39.13/5.42  = { by axiom 1 (ifeq_axiom) R->L }
% 39.13/5.42    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 39.13/5.42  = { by axiom 5 (cn_1) R->L }
% 39.13/5.42    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 39.13/5.42  = { by axiom 1 (ifeq_axiom) R->L }
% 39.13/5.42    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 39.13/5.43  = { by axiom 2 (cn_3) R->L }
% 39.13/5.43    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))))), true, ifeq(is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))))), true), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 39.13/5.43  = { by axiom 4 (condensed_detachment) }
% 39.13/5.43    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 39.13/5.43  = { by axiom 1 (ifeq_axiom) }
% 39.13/5.43    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x))))), true, ifeq(is_a_theorem(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(x), y), implies(implies(y, x), x)), implies(implies(y, x), x)), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true), true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 39.13/5.43  = { by axiom 4 (condensed_detachment) }
% 39.13/5.43    ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 39.13/5.43  = { by axiom 1 (ifeq_axiom) }
% 39.13/5.43    ifeq(is_a_theorem(implies(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x))), implies(implies(not(x), y), implies(implies(y, x), x)))), true, ifeq(is_a_theorem(implies(implies(not(x), y), implies(implies(not(x), y), implies(implies(y, x), x)))), true, is_a_theorem(implies(implies(not(x), y), implies(implies(y, x), x))), true), true)
% 39.13/5.43  = { by axiom 4 (condensed_detachment) }
% 39.13/5.43    true
% 39.13/5.43  % SZS output end Proof
% 39.13/5.43  
% 39.13/5.43  RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------