%   ORIGINAL: 'h4/thm/pred_set/IN_CROSS_'
% 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: !P Q x. 'h4/const/bool/IN' x ('h4/const/pred_set/CROSS' P Q) <=> 'h4/const/bool/IN' ('h4/const/pair/FST' x) P /\ 'h4/const/bool/IN' ('h4/const/pair/SND' x) Q
%   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: !P Q x. 'h4/const/bool/IN' x ('h4/const/pred_set/CROSS' P Q) <=> 'h4/const/bool/IN' ('h4/const/pair/FST' x) P /\ 'h4/const/bool/IN' ('h4/const/pair/SND' x) Q
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_3f21021,V_3f21017]: ![F, G]: (![X]: s(V_3f21017,happ(s(fun(V_3f21021,V_3f21017),F),s(V_3f21021,X))) = s(V_3f21017,happ(s(fun(V_3f21021,V_3f21017),G),s(V_3f21021,X))) => s(fun(V_3f21021,V_3f21017),F) = s(fun(V_3f21021,V_3f21017),G))).
fof('h4/thm/pred_set/IN_CROSS_', conjecture, ![A,B]: ![P, Q, X]: (p(s(bool,'h4/const/bool/IN'(s('h4/type/pair/prod'(A,B),X),s(fun('h4/type/pair/prod'(A,B),bool),'h4/const/pred_set/CROSS'(s(fun(A,bool),P),s(fun(B,bool),Q)))))) <=> (p(s(bool,'h4/const/bool/IN'(s(A,'h4/const/pair/FST'(s('h4/type/pair/prod'(A,B),X))),s(fun(A,bool),P)))) & p(s(bool,'h4/const/bool/IN'(s(B,'h4/const/pair/SND'(s('h4/type/pair/prod'(A,B),X))),s(fun(B,bool),Q))))))).
