%   ORIGINAL: 'h4/thm/pred_set/GSPEC_EQ_'
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Goal: !y. 'h4/const/pred_set/GSPEC' (\x. 'h4/const/pair/,' x (x = y)) = 'h4/const/pred_set/INSERT' y 'h4/const/pred_set/EMPTY'
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !_0. (!y x. ?v. (v <=> x = y) /\ happ (happ _0 y) x = 'h4/const/pair/,' x v) ==> (!y. 'h4/const/pred_set/GSPEC' (happ _0 y) = 'h4/const/pred_set/INSERT' y 'h4/const/pred_set/EMPTY')
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f23573,V_3f23569]: ![F, G]: (![X]: s(V_3f23569,happ(s(fun(V_3f23573,V_3f23569),F),s(V_3f23573,X))) = s(V_3f23569,happ(s(fun(V_3f23573,V_3f23569),G),s(V_3f23573,X))) => s(fun(V_3f23573,V_3f23569),F) = s(fun(V_3f23573,V_3f23569),G))).
fof('h4/thm/pred_set/GSPEC_EQ_', conjecture, ![A]: ![V__0]: (![Y, X]: ?[V]: ((p(s(bool,V)) <=> s(A,X) = s(A,Y)) & s('h4/type/pair/prod'(A,bool),happ(s(fun(A,'h4/type/pair/prod'(A,bool)),happ(s(fun(A,fun(A,'h4/type/pair/prod'(A,bool))),V__0),s(A,Y))),s(A,X))) = s('h4/type/pair/prod'(A,bool),'h4/const/pair/,'(s(A,X),s(bool,V)))) => ![Y]: s(fun(A,bool),'h4/const/pred_set/GSPEC'(s(fun(A,'h4/type/pair/prod'(A,bool)),happ(s(fun(A,fun(A,'h4/type/pair/prod'(A,bool))),V__0),s(A,Y))))) = s(fun(A,bool),'h4/const/pred_set/INSERT'(s(A,Y),s(fun(A,bool),'h4/const/pred_set/EMPTY'))))).
