Twee---2.7.UNS-Prf.s

View TPTP
Problem Process solution in
SystemOnTSTP Download .tgz
%------------------------------------------------------------------------------
% File     : Twee---2.7
% Problem  : LCL405-1 : TPTP v9.3.1. Released v2.3.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p

% Computer : n011.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 11:49:47 AM UTC 2026

% Result   : Unsatisfiable 27.46s 3.95s
% Output   : Proof 40.77s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03  % Problem  : LCL405-1 : TPTP v9.3.1. Released v2.3.0.
% 0.00/0.04  % Command  : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.10/0.36  % Computer : n011.cluster.edu
% 0.10/0.36  % Model    : x86_64 x86_64
% 0.10/0.36  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.36  % Memory   : 8046.5625MB
% 0.10/0.36  % OS       : Linux 6.8.0-71-generic
% 0.10/0.36  % CPULimit : 300
% 0.10/0.36  % WCLimit  : 300
% 0.10/0.36  % DateTime : Sun Sep 27 15:40:30 UTC 2026
% 0.10/0.36  % CPUTime  : 
% 0.10/0.37  Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 27.46/3.95  Command-line arguments: --no-flatten-goal
% 27.46/3.95  
% 27.46/3.95  % SZS status Unsatisfiable
% 27.46/3.95  
% 34.75/4.95  % SZS output start Proof
% 34.75/4.95  Axiom 1 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 34.75/4.95  Axiom 2 (cn_3): is_a_theorem(implies(X, implies(not(X), Y))) = true.
% 34.75/4.95  Axiom 3 (cn_2): is_a_theorem(implies(implies(not(X), X), X)) = true.
% 34.75/4.95  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.
% 34.75/4.98  Axiom 5 (cn_1): is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
% 34.75/4.98  
% 34.75/4.98  Goal 1 (prove_cn_71): is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))) = true.
% 34.75/4.98  Proof:
% 34.75/4.98    is_a_theorem(not(implies(implies(x, x), not(implies(y, y)))))
% 34.75/4.98  = { by axiom 1 (ifeq_axiom) R->L }
% 34.75/4.98    ifeq(true, true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/4.98  = { by axiom 4 (condensed_detachment) R->L }
% 34.75/4.98    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(is_a_theorem(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/4.98  = { by axiom 1 (ifeq_axiom) R->L }
% 34.75/4.98    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/4.98  = { by axiom 4 (condensed_detachment) R->L }
% 34.75/4.98    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(y), y), y), implies(y, y))), true, ifeq(is_a_theorem(implies(implies(not(y), y), y)), true, is_a_theorem(implies(y, y)), true), true), true, is_a_theorem(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/4.98  = { by axiom 3 (cn_2) }
% 34.75/4.99    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(y), y), y), implies(y, y))), true, ifeq(true, true, is_a_theorem(implies(y, y)), true), true), true, is_a_theorem(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/4.99  = { by axiom 1 (ifeq_axiom) }
% 34.75/4.99    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(y), y), y), implies(y, y))), true, is_a_theorem(implies(y, y)), true), true, is_a_theorem(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/4.99  = { by axiom 1 (ifeq_axiom) R->L }
% 34.75/4.99    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(y), y), y), implies(y, y))), true), true, is_a_theorem(implies(y, y)), true), true, is_a_theorem(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/4.99  = { by axiom 5 (cn_1) R->L }
% 34.75/4.99    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(y, implies(not(y), y)), implies(implies(implies(not(y), y), y), implies(y, y)))), true, is_a_theorem(implies(implies(implies(not(y), y), y), implies(y, y))), true), true, is_a_theorem(implies(y, y)), true), true, is_a_theorem(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/4.99  = { by axiom 1 (ifeq_axiom) R->L }
% 34.75/4.99    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(y, implies(not(y), y)), implies(implies(implies(not(y), y), y), implies(y, y)))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(y), y), y), implies(y, y))), true), true), true, is_a_theorem(implies(y, y)), true), true, is_a_theorem(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/4.99  = { by axiom 2 (cn_3) R->L }
% 34.75/4.99    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(y, implies(not(y), y)), implies(implies(implies(not(y), y), y), implies(y, y)))), true, ifeq(is_a_theorem(implies(y, implies(not(y), y))), true, is_a_theorem(implies(implies(implies(not(y), y), y), implies(y, y))), true), true), true, is_a_theorem(implies(y, y)), true), true, is_a_theorem(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/4.99  = { by axiom 4 (condensed_detachment) }
% 34.75/4.99    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(y, y)), true), true, is_a_theorem(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/4.99  = { by axiom 1 (ifeq_axiom) }
% 34.75/4.99    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(is_a_theorem(implies(y, y)), true, is_a_theorem(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.00  = { by axiom 1 (ifeq_axiom) R->L }
% 34.75/5.00    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(y, y)), true, is_a_theorem(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.00  = { by axiom 2 (cn_3) R->L }
% 34.75/5.00    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(y, y), implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(is_a_theorem(implies(y, y)), true, is_a_theorem(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.00  = { by axiom 4 (condensed_detachment) }
% 34.75/5.00    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.00  = { by axiom 1 (ifeq_axiom) }
% 34.75/5.00    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.00  = { by axiom 1 (ifeq_axiom) R->L }
% 34.75/5.00    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.00  = { by axiom 4 (condensed_detachment) R->L }
% 34.75/5.00    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y)))), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.00  = { by axiom 3 (cn_2) }
% 34.75/5.00    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.00  = { by axiom 1 (ifeq_axiom) }
% 34.75/5.00    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.00  = { by axiom 1 (ifeq_axiom) R->L }
% 34.75/5.00    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.01  = { by axiom 4 (condensed_detachment) R->L }
% 34.75/5.01    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(y, y))), implies(x, x)), implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y)))))), true, ifeq(is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.01  = { by axiom 5 (cn_1) }
% 34.75/5.01    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.01  = { by axiom 1 (ifeq_axiom) }
% 34.75/5.01    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.01  = { by axiom 1 (ifeq_axiom) R->L }
% 34.75/5.01    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.01  = { by axiom 4 (condensed_detachment) R->L }
% 34.75/5.01    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(y, y))), implies(not(implies(x, x)), implies(x, x))), implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(not(not(implies(y, y))), implies(x, x))))), true, ifeq(is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.01  = { by axiom 5 (cn_1) }
% 34.75/5.01    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.02  = { by axiom 1 (ifeq_axiom) }
% 34.75/5.02    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.02  = { by axiom 1 (ifeq_axiom) R->L }
% 34.75/5.02    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.02  = { by axiom 4 (condensed_detachment) R->L }
% 34.75/5.02    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(x, x)), not(not(not(implies(y, y))))), implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x))))), true, ifeq(is_a_theorem(implies(not(implies(x, x)), not(not(not(implies(y, y)))))), true, is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.02  = { by axiom 5 (cn_1) }
% 34.75/5.03    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(implies(x, x)), not(not(not(implies(y, y)))))), true, is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.03  = { by axiom 1 (ifeq_axiom) }
% 34.75/5.03    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(implies(x, x)), not(not(not(implies(y, y)))))), true, is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.03  = { by axiom 1 (ifeq_axiom) R->L }
% 34.75/5.03    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(x, x)), not(not(not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.03  = { by axiom 4 (condensed_detachment) R->L }
% 34.75/5.03    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(not(not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.04  = { by axiom 3 (cn_2) }
% 34.75/5.04    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(not(not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.04  = { by axiom 1 (ifeq_axiom) }
% 34.75/5.04    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(not(not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.04  = { by axiom 1 (ifeq_axiom) R->L }
% 34.75/5.04    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(not(not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.05  = { by axiom 5 (cn_1) R->L }
% 34.75/5.05    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(not(not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.05  = { by axiom 1 (ifeq_axiom) R->L }
% 34.75/5.05    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(not(not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.05  = { by axiom 2 (cn_3) R->L }
% 34.75/5.05    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(not(not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.05  = { by axiom 4 (condensed_detachment) }
% 34.75/5.06    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(not(not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.06  = { by axiom 1 (ifeq_axiom) }
% 34.75/5.06    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(not(not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.06  = { by axiom 1 (ifeq_axiom) R->L }
% 34.75/5.06    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(not(not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 34.75/5.06  = { by axiom 2 (cn_3) R->L }
% 34.75/5.06    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(x, x), implies(not(implies(x, x)), not(not(not(implies(y, y))))))), true, ifeq(is_a_theorem(implies(x, x)), true, is_a_theorem(implies(not(implies(x, x)), not(not(not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.07  = { by axiom 4 (condensed_detachment) }
% 36.51/5.07    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.07  = { by axiom 1 (ifeq_axiom) }
% 36.51/5.07    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.07  = { by axiom 1 (ifeq_axiom) R->L }
% 36.51/5.07    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.07  = { by axiom 4 (condensed_detachment) R->L }
% 36.51/5.08    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(y, y))), implies(not(not(not(implies(y, y)))), implies(x, x))), implies(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x))), implies(not(not(implies(y, y))), implies(not(implies(x, x)), implies(x, x)))))), true, ifeq(is_a_theorem(implies(not(not(implies(y, y))), implies(not(not(not(implies(y, y)))), implies(x, x)))), true, is_a_theorem(implies(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x))), implies(not(not(implies(y, y))), implies(not(implies(x, x)), implies(x, x))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.08  = { by axiom 2 (cn_3) }
% 36.51/5.08    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(y, y))), implies(not(not(not(implies(y, y)))), implies(x, x))), implies(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x))), implies(not(not(implies(y, y))), implies(not(implies(x, x)), implies(x, x)))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x))), implies(not(not(implies(y, y))), implies(not(implies(x, x)), implies(x, x))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.08  = { by axiom 1 (ifeq_axiom) }
% 36.51/5.08    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(y, y))), implies(not(not(not(implies(y, y)))), implies(x, x))), implies(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x))), implies(not(not(implies(y, y))), implies(not(implies(x, x)), implies(x, x)))))), true, is_a_theorem(implies(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x))), implies(not(not(implies(y, y))), implies(not(implies(x, x)), implies(x, x))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.08  = { by axiom 5 (cn_1) }
% 36.51/5.08    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x))), implies(not(not(implies(y, y))), implies(not(implies(x, x)), implies(x, x))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.08  = { by axiom 1 (ifeq_axiom) }
% 36.51/5.09    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x))), implies(not(not(implies(y, y))), implies(not(implies(x, x)), implies(x, x))))), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(y, y)))), implies(x, x)), implies(not(implies(x, x)), implies(x, x)))), true, is_a_theorem(implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.09  = { by axiom 4 (condensed_detachment) }
% 36.51/5.09    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(not(not(implies(y, y))), implies(x, x)))), true), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.09  = { by axiom 1 (ifeq_axiom) }
% 36.51/5.09    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(not(not(implies(y, y))), implies(x, x)))), true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.09  = { by axiom 1 (ifeq_axiom) R->L }
% 36.51/5.09    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(not(not(implies(y, y))), implies(x, x)))), true, ifeq(true, true, is_a_theorem(implies(not(not(implies(y, y))), implies(x, x))), true), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.09  = { by axiom 3 (cn_2) R->L }
% 36.51/5.09    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(x, x)), implies(x, x)), implies(x, x)), implies(not(not(implies(y, 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(not(not(implies(y, y))), implies(x, x))), true), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.09  = { by axiom 4 (condensed_detachment) }
% 36.51/5.09    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.09  = { by axiom 1 (ifeq_axiom) }
% 36.51/5.10    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.10  = { by axiom 1 (ifeq_axiom) R->L }
% 36.51/5.10    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.10  = { by axiom 5 (cn_1) R->L }
% 36.51/5.10    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y)))), implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))))), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), implies(not(not(implies(y, y))), not(implies(y, y))))), true, is_a_theorem(implies(implies(implies(not(not(implies(y, y))), not(implies(y, y))), not(implies(y, y))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))))), true), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.10  = { by axiom 4 (condensed_detachment) }
% 36.51/5.10    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.10  = { by axiom 1 (ifeq_axiom) }
% 36.51/5.10    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.10  = { by axiom 1 (ifeq_axiom) R->L }
% 36.51/5.10    ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.10  = { by axiom 5 (cn_1) R->L }
% 36.51/5.10    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y))), implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))))), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(y, y)))), true, is_a_theorem(implies(implies(not(implies(y, y)), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.10  = { by axiom 4 (condensed_detachment) }
% 36.51/5.10    ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.10  = { by axiom 1 (ifeq_axiom) }
% 36.51/5.10    ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true)
% 36.51/5.10  = { by axiom 1 (ifeq_axiom) R->L }
% 36.51/5.10    ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.10  = { by axiom 4 (condensed_detachment) R->L }
% 36.51/5.11    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.11  = { by axiom 1 (ifeq_axiom) R->L }
% 36.51/5.11    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.11  = { by axiom 4 (condensed_detachment) R->L }
% 36.51/5.11    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y)))))))), true, ifeq(is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.11  = { by axiom 2 (cn_3) }
% 36.51/5.11    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y)))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.11  = { by axiom 1 (ifeq_axiom) }
% 36.51/5.12    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y)))))))), true, is_a_theorem(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.12  = { by axiom 5 (cn_1) }
% 36.51/5.12    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.12  = { by axiom 1 (ifeq_axiom) }
% 36.51/5.12    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.12  = { by axiom 1 (ifeq_axiom) R->L }
% 36.51/5.12    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(true, true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.12  = { by axiom 3 (cn_2) R->L }
% 36.51/5.12    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.12  = { by axiom 4 (condensed_detachment) }
% 36.51/5.12    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.13  = { by axiom 1 (ifeq_axiom) }
% 36.51/5.13    ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.13  = { by axiom 1 (ifeq_axiom) R->L }
% 36.51/5.13    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.13  = { by axiom 4 (condensed_detachment) R->L }
% 36.51/5.14    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.14  = { by axiom 1 (ifeq_axiom) R->L }
% 36.51/5.14    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.14  = { by axiom 4 (condensed_detachment) R->L }
% 36.51/5.14    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y)))))))), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.15  = { by axiom 3 (cn_2) }
% 36.51/5.15    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y)))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.15  = { by axiom 1 (ifeq_axiom) }
% 36.51/5.15    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y)))))))), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.15  = { by axiom 5 (cn_1) }
% 36.51/5.15    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.16  = { by axiom 1 (ifeq_axiom) }
% 36.51/5.16    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.16  = { by axiom 1 (ifeq_axiom) R->L }
% 36.51/5.16    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 36.51/5.16  = { by axiom 5 (cn_1) R->L }
% 36.51/5.16    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.28/5.16  = { by axiom 4 (condensed_detachment) }
% 37.28/5.17    ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.28/5.17  = { by axiom 1 (ifeq_axiom) }
% 37.28/5.17    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.28/5.17  = { by axiom 1 (ifeq_axiom) R->L }
% 37.28/5.17    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.28/5.18  = { by axiom 4 (condensed_detachment) R->L }
% 37.28/5.18    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.28/5.18  = { by axiom 1 (ifeq_axiom) R->L }
% 37.28/5.18    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.28/5.19  = { by axiom 4 (condensed_detachment) R->L }
% 37.28/5.19    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.28/5.20  = { by axiom 1 (ifeq_axiom) R->L }
% 37.28/5.20    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.59/5.21  = { by axiom 5 (cn_1) R->L }
% 37.59/5.21    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.59/5.21  = { by axiom 1 (ifeq_axiom) R->L }
% 37.59/5.22    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.59/5.22  = { by axiom 2 (cn_3) R->L }
% 37.59/5.23    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true, ifeq(is_a_theorem(implies(implies(x, x), implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))))), true, is_a_theorem(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.59/5.23  = { by axiom 4 (condensed_detachment) }
% 37.59/5.23    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true, ifeq(true, true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.59/5.24  = { by axiom 1 (ifeq_axiom) }
% 37.59/5.24    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.59/5.25  = { by axiom 1 (ifeq_axiom) R->L }
% 37.59/5.25    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.59/5.26  = { by axiom 4 (condensed_detachment) R->L }
% 37.59/5.26    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.59/5.27  = { by axiom 1 (ifeq_axiom) R->L }
% 37.59/5.27    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.59/5.28  = { by axiom 5 (cn_1) R->L }
% 37.59/5.28    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x)))), implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))))), true, is_a_theorem(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.59/5.29  = { by axiom 1 (ifeq_axiom) R->L }
% 37.59/5.30    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x)))), implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 37.59/5.30  = { by axiom 2 (cn_3) R->L }
% 38.37/5.31    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x)))), implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))))), true, ifeq(is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x))))), true, is_a_theorem(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 38.37/5.31  = { by axiom 4 (condensed_detachment) }
% 38.37/5.31    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 38.37/5.32  = { by axiom 1 (ifeq_axiom) }
% 38.37/5.32    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(x, x))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 38.37/5.32  = { by axiom 5 (cn_1) }
% 38.37/5.33    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 38.37/5.33  = { by axiom 1 (ifeq_axiom) }
% 38.37/5.33    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 38.37/5.34  = { by axiom 1 (ifeq_axiom) R->L }
% 38.37/5.34    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 38.37/5.34  = { by axiom 5 (cn_1) R->L }
% 38.37/5.35    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))))), true, ifeq(is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(x, x)), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))))), true), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), 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(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 38.37/5.35  = { by axiom 1 (ifeq_axiom) }
% 38.37/5.35    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 38.37/5.35  = { 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(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 38.37/5.36  = { by axiom 5 (cn_1) R->L }
% 38.37/5.36    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))))), implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))))), true, ifeq(is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))))), true, is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 38.37/5.36  = { by axiom 4 (condensed_detachment) }
% 38.37/5.36    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 38.37/5.37  = { by axiom 1 (ifeq_axiom) }
% 38.37/5.37    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 38.37/5.37  = { by axiom 1 (ifeq_axiom) R->L }
% 38.37/5.37    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 38.37/5.37  = { by axiom 4 (condensed_detachment) R->L }
% 38.37/5.38    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y)))), implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true, ifeq(is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), 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, ifeq(is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), 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(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 38.37/5.39  = { by axiom 1 (ifeq_axiom) R->L }
% 38.37/5.39    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 38.37/5.39  = { by axiom 4 (condensed_detachment) R->L }
% 38.37/5.40    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y)))))), true, ifeq(is_a_theorem(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true), true), true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 38.37/5.40  = { by axiom 3 (cn_2) }
% 38.37/5.40    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y)))))), true, ifeq(true, true, is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true), true), true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.18/5.41  = { by axiom 1 (ifeq_axiom) }
% 39.18/5.41    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y)))))), true, is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true), true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.18/5.41  = { by axiom 5 (cn_1) }
% 39.18/5.41    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true), true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.18/5.42  = { by axiom 1 (ifeq_axiom) }
% 39.18/5.42    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.18/5.43  = { by axiom 1 (ifeq_axiom) R->L }
% 39.18/5.43    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.18/5.44  = { by axiom 4 (condensed_detachment) R->L }
% 39.18/5.44    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y)))), implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))))), true, ifeq(is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))))), true, is_a_theorem(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.18/5.44  = { by axiom 2 (cn_3) }
% 39.18/5.45    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y)))), implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.18/5.45  = { by axiom 1 (ifeq_axiom) }
% 39.18/5.45    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y)))), implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.18/5.45  = { by axiom 5 (cn_1) }
% 39.18/5.46    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.18/5.46  = { by axiom 1 (ifeq_axiom) }
% 39.18/5.47    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y)))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(y, y))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true, is_a_theorem(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.18/5.47  = { by axiom 4 (condensed_detachment) }
% 39.18/5.47    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.18/5.47  = { by axiom 1 (ifeq_axiom) }
% 39.18/5.47    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.18/5.48  = { by axiom 1 (ifeq_axiom) R->L }
% 39.18/5.48    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.18/5.48  = { by axiom 4 (condensed_detachment) R->L }
% 39.18/5.49    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))))), true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.18/5.49  = { by axiom 5 (cn_1) }
% 39.18/5.49    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.18/5.50  = { by axiom 1 (ifeq_axiom) }
% 39.18/5.50    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.18/5.50  = { by axiom 5 (cn_1) }
% 39.96/5.50    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.50  = { by axiom 1 (ifeq_axiom) }
% 39.96/5.50    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y)))))), not(implies(y, y))), implies(implies(x, x), not(implies(y, y)))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(implies(implies(x, x), implies(not(not(not(implies(implies(x, x), not(implies(y, y)))))), not(not(implies(implies(x, x), not(implies(y, y))))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true), true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.51  = { by axiom 4 (condensed_detachment) }
% 39.96/5.51    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.51  = { by axiom 1 (ifeq_axiom) }
% 39.96/5.51    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.51  = { by axiom 1 (ifeq_axiom) R->L }
% 39.96/5.51    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.51  = { by axiom 5 (cn_1) R->L }
% 39.96/5.51    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true, ifeq(is_a_theorem(implies(not(not(implies(implies(x, x), not(implies(y, y))))), implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.51  = { by axiom 4 (condensed_detachment) }
% 39.96/5.52    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.52  = { by axiom 1 (ifeq_axiom) }
% 39.96/5.52    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.52  = { by axiom 1 (ifeq_axiom) R->L }
% 39.96/5.53    ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.53  = { by axiom 4 (condensed_detachment) R->L }
% 39.96/5.53    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true, is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.53  = { by axiom 1 (ifeq_axiom) R->L }
% 39.96/5.54    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true, is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.54  = { by axiom 5 (cn_1) R->L }
% 39.96/5.55    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true, is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.55  = { by axiom 1 (ifeq_axiom) R->L }
% 39.96/5.55    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true, is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.55  = { by axiom 5 (cn_1) R->L }
% 39.96/5.56    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true, is_a_theorem(implies(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true, is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.56  = { by axiom 4 (condensed_detachment) }
% 39.96/5.56    ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true, is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.57  = { by axiom 1 (ifeq_axiom) }
% 39.96/5.57    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true, is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.57  = { by axiom 1 (ifeq_axiom) R->L }
% 39.96/5.58    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.58  = { by axiom 4 (condensed_detachment) R->L }
% 39.96/5.58    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.58  = { by axiom 3 (cn_2) }
% 39.96/5.58    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true, ifeq(true, true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.59  = { by axiom 1 (ifeq_axiom) }
% 39.96/5.59    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.59  = { by axiom 5 (cn_1) }
% 39.96/5.60    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.60  = { by axiom 1 (ifeq_axiom) }
% 39.96/5.60    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 39.96/5.60  = { by axiom 1 (ifeq_axiom) R->L }
% 40.77/5.60    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 40.77/5.60  = { by axiom 5 (cn_1) R->L }
% 40.77/5.61    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))))), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(implies(implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 40.77/5.61  = { by axiom 4 (condensed_detachment) }
% 40.77/5.61    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 40.77/5.61  = { by axiom 1 (ifeq_axiom) }
% 40.77/5.61    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), implies(not(not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true, is_a_theorem(implies(implies(not(implies(implies(x, x), not(implies(y, y)))), not(implies(implies(x, x), not(implies(y, y))))), implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 40.77/5.61  = { by axiom 4 (condensed_detachment) }
% 40.77/5.61    ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 40.77/5.61  = { by axiom 1 (ifeq_axiom) }
% 40.77/5.61    ifeq(is_a_theorem(implies(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y))))), not(implies(implies(x, x), not(implies(y, y)))))), true, ifeq(is_a_theorem(implies(implies(implies(x, x), not(implies(y, y))), not(implies(implies(x, x), not(implies(y, y)))))), true, is_a_theorem(not(implies(implies(x, x), not(implies(y, y))))), true), true)
% 40.77/5.61  = { by axiom 4 (condensed_detachment) }
% 40.77/5.61    true
% 40.77/5.61  % SZS output end Proof
% 40.77/5.61  
% 40.77/5.61  RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------