%   ORIGINAL: 'h4/thm/pred_set/GSPECIFICATION_applied_'
% 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: !f v. 'h4/const/pred_set/GSPEC' f v <=> (?x. 'h4/const/pair/,' v T = f x)
%   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: !f v. 'h4/const/pred_set/GSPEC' f v <=> (?x. 'h4/const/pair/,' v T = happ f x)
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_3f14769,V_3f14765]: ![F, G]: (![X]: s(V_3f14765,happ(s(fun(V_3f14769,V_3f14765),F),s(V_3f14769,X))) = s(V_3f14765,happ(s(fun(V_3f14769,V_3f14765),G),s(V_3f14769,X))) => s(fun(V_3f14769,V_3f14765),F) = s(fun(V_3f14769,V_3f14765),G))).
fof('h4/thm/pred_set/GSPECIFICATION_applied_', conjecture, ![A,B]: ![F, V]: (p(s(bool,'h4/const/pred_set/GSPEC'(s(fun(B,'h4/type/pair/prod'(A,bool)),F),s(A,V)))) <=> ?[X]: s('h4/type/pair/prod'(A,bool),'h4/const/pair/,'(s(A,V),s(bool,'T'))) = s('h4/type/pair/prod'(A,bool),happ(s(fun(B,'h4/type/pair/prod'(A,bool)),F),s(B,X))))).
