%   ORIGINAL: 'h4/thm/bool/RIGHT_AND_FORALL_THM_'
% 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. P /\ (!x. Q x) <=> (!x. P /\ Q 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: !P Q. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q 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_3f33128,V_3f33124]: ![F, G]: (![X]: s(V_3f33124,happ(s(fun(V_3f33128,V_3f33124),F),s(V_3f33128,X))) = s(V_3f33124,happ(s(fun(V_3f33128,V_3f33124),G),s(V_3f33128,X))) => s(fun(V_3f33128,V_3f33124),F) = s(fun(V_3f33128,V_3f33124),G))).
fof('h4/thm/bool/RIGHT_AND_FORALL_THM_', conjecture, ![A]: ![P, Q]: ((p(s(bool,P)) & ![X]: p(s(bool,happ(s(fun(A,bool),Q),s(A,X))))) <=> ![X]: (p(s(bool,P)) & p(s(bool,happ(s(fun(A,bool),Q),s(A,X))))))).
