%   ORIGINAL: 'h4/thm/pair/ELIM_PFORALL_EVAL_'
% 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. $forall ('h4/const/pair/UNCURRY' (\x. P x)) <=> (!x. $forall (P 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: !_0. (!P x. happ (happ _0 P) x = happ P x) ==> (!P. $forall ('h4/const/pair/UNCURRY' (happ _0 P)) <=> (!x. $forall (happ P x)))
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f61361,V_3f61357]: ![F, G]: (![X]: s(V_3f61357,happ(s(fun(V_3f61361,V_3f61357),F),s(V_3f61361,X))) = s(V_3f61357,happ(s(fun(V_3f61361,V_3f61357),G),s(V_3f61361,X))) => s(fun(V_3f61361,V_3f61357),F) = s(fun(V_3f61361,V_3f61357),G))).
fof('h4/thm/pair/ELIM_PFORALL_EVAL_', conjecture, ![B,A]: ![V__0]: (![P, X]: s(fun(B,bool),happ(s(fun(A,fun(B,bool)),happ(s(fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),V__0),s(fun(A,fun(B,bool)),P))),s(A,X))) = s(fun(B,bool),happ(s(fun(A,fun(B,bool)),P),s(A,X))) => ![P]: (p(s(bool,'$forall'(s(fun('h4/type/pair/prod'(A,B),bool),'h4/const/pair/UNCURRY'(s(fun(A,fun(B,bool)),happ(s(fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),V__0),s(fun(A,fun(B,bool)),P)))))))) <=> ![X]: p(s(bool,'$forall'(s(fun(B,bool),happ(s(fun(A,fun(B,bool)),P),s(A,X))))))))).
