%   ORIGINAL: 'h4/thm/res_quan/RES_EXISTS_NULL_'
% 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 m. 'h4/const/bool/RES_EXISTS' p (\x. m) <=> ~(p = 'h4/const/pred_set/EMPTY') /\ m
%   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. (!m x. happ (happ _0 m) x <=> m) ==> (!p m. 'h4/const/bool/RES_EXISTS' p (happ _0 m) <=> ~(p = 'h4/const/pred_set/EMPTY') /\ m)
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_3f31128,V_3f31124]: ![F, G]: (![X]: s(V_3f31124,happ(s(fun(V_3f31128,V_3f31124),F),s(V_3f31128,X))) = s(V_3f31124,happ(s(fun(V_3f31128,V_3f31124),G),s(V_3f31128,X))) => s(fun(V_3f31128,V_3f31124),F) = s(fun(V_3f31128,V_3f31124),G))).
fof('h4/thm/res_quan/RES_EXISTS_NULL_', conjecture, ![A]: ![V__0]: (![M, X]: s(bool,happ(s(fun(A,bool),happ(s(fun(bool,fun(A,bool)),V__0),s(bool,M))),s(A,X))) = s(bool,M) => ![P, M]: (p(s(bool,'h4/const/bool/RES_EXISTS'(s(fun(A,bool),P),s(fun(A,bool),happ(s(fun(bool,fun(A,bool)),V__0),s(bool,M)))))) <=> (~ (s(fun(A,bool),P) = s(fun(A,bool),'h4/const/pred_set/EMPTY')) & p(s(bool,M)))))).
