%   ORIGINAL: 'h4/thm/pred_set/GSPEC_ID_'
% 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 ('h4/const/bool/IN' x y)) = y
%   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. happ (happ _0 y) x = 'h4/const/pair/,' x ('h4/const/bool/IN' x y)) ==> (!y. 'h4/const/pred_set/GSPEC' (happ _0 y) = y)
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_3f23548,V_3f23544]: ![F, G]: (![X]: s(V_3f23544,happ(s(fun(V_3f23548,V_3f23544),F),s(V_3f23548,X))) = s(V_3f23544,happ(s(fun(V_3f23548,V_3f23544),G),s(V_3f23548,X))) => s(fun(V_3f23548,V_3f23544),F) = s(fun(V_3f23548,V_3f23544),G))).
fof('h4/thm/pred_set/GSPEC_ID_', conjecture, ![A]: ![V__0]: (![Y, X]: s('h4/type/pair/prod'(A,bool),happ(s(fun(A,'h4/type/pair/prod'(A,bool)),happ(s(fun(fun(A,bool),fun(A,'h4/type/pair/prod'(A,bool))),V__0),s(fun(A,bool),Y))),s(A,X))) = s('h4/type/pair/prod'(A,bool),'h4/const/pair/,'(s(A,X),s(bool,'h4/const/bool/IN'(s(A,X),s(fun(A,bool),Y))))) => ![Y]: s(fun(A,bool),'h4/const/pred_set/GSPEC'(s(fun(A,'h4/type/pair/prod'(A,bool)),happ(s(fun(fun(A,bool),fun(A,'h4/type/pair/prod'(A,bool))),V__0),s(fun(A,bool),Y))))) = s(fun(A,bool),Y))).
