%   ORIGINAL: 'h4/thm/res_quan/RES_SELECT_'
% 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 f. 'h4/const/bool/RES_SELECT' P f = 'h4/const/min/@' (\x. 'h4/const/bool/IN' x P /\ 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: !_0. (!P f x. happ (happ (happ _0 P) f) x <=> 'h4/const/bool/IN' x P /\ happ f x) ==> (!P f. 'h4/const/bool/RES_SELECT' P f = 'h4/const/min/@' (happ (happ _0 P) f))
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_3f30753,V_3f30749]: ![F, G]: (![X]: s(V_3f30749,happ(s(fun(V_3f30753,V_3f30749),F),s(V_3f30753,X))) = s(V_3f30749,happ(s(fun(V_3f30753,V_3f30749),G),s(V_3f30753,X))) => s(fun(V_3f30753,V_3f30749),F) = s(fun(V_3f30753,V_3f30749),G))).
fof('h4/thm/res_quan/RES_SELECT_', conjecture, ![A]: ![V__0]: (![P, F, X]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun(fun(A,bool),fun(A,bool)),happ(s(fun(fun(A,bool),fun(fun(A,bool),fun(A,bool))),V__0),s(fun(A,bool),P))),s(fun(A,bool),F))),s(A,X)))) <=> (p(s(bool,'h4/const/bool/IN'(s(A,X),s(fun(A,bool),P)))) & p(s(bool,happ(s(fun(A,bool),F),s(A,X)))))) => ![P, F]: s(A,'h4/const/bool/RES_SELECT'(s(fun(A,bool),P),s(fun(A,bool),F))) = s(A,'h4/const/min/@'(s(fun(A,bool),happ(s(fun(fun(A,bool),fun(A,bool)),happ(s(fun(fun(A,bool),fun(fun(A,bool),fun(A,bool))),V__0),s(fun(A,bool),P))),s(fun(A,bool),F))))))).
