%   ORIGINAL: 'h4/thm/res_quan/RES_FORALL_DISJ_DIST_'
% 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 R. 'h4/const/bool/RES_FORALL' (\j. P j \/ Q j) (\i. R i) <=> 'h4/const/bool/RES_FORALL' P (\i. R i) /\ 'h4/const/bool/RES_FORALL' Q (\i. R i)
%   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: !_1. (!R i. happ (happ _1 R) i <=> happ R i) ==> (!_0. (!P Q j. happ (happ (happ _0 P) Q) j <=> happ P j \/ happ Q j) ==> (!P Q R. 'h4/const/bool/RES_FORALL' (happ (happ _0 P) Q) (happ _1 R) <=> 'h4/const/bool/RES_FORALL' P (happ _1 R) /\ 'h4/const/bool/RES_FORALL' Q (happ _1 R)))
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_3f30803,V_3f30799]: ![F, G]: (![X]: s(V_3f30799,happ(s(fun(V_3f30803,V_3f30799),F),s(V_3f30803,X))) = s(V_3f30799,happ(s(fun(V_3f30803,V_3f30799),G),s(V_3f30803,X))) => s(fun(V_3f30803,V_3f30799),F) = s(fun(V_3f30803,V_3f30799),G))).
fof('h4/thm/res_quan/RES_FORALL_DISJ_DIST_', conjecture, ![A]: ![V__1]: (![R, I]: s(bool,happ(s(fun(A,bool),happ(s(fun(fun(A,bool),fun(A,bool)),V__1),s(fun(A,bool),R))),s(A,I))) = s(bool,happ(s(fun(A,bool),R),s(A,I))) => ![V__0]: (![P, Q, J]: (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),Q))),s(A,J)))) <=> (p(s(bool,happ(s(fun(A,bool),P),s(A,J)))) | p(s(bool,happ(s(fun(A,bool),Q),s(A,J)))))) => ![P, Q, R]: (p(s(bool,'h4/const/bool/RES_FORALL'(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),Q))),s(fun(A,bool),happ(s(fun(fun(A,bool),fun(A,bool)),V__1),s(fun(A,bool),R)))))) <=> (p(s(bool,'h4/const/bool/RES_FORALL'(s(fun(A,bool),P),s(fun(A,bool),happ(s(fun(fun(A,bool),fun(A,bool)),V__1),s(fun(A,bool),R)))))) & p(s(bool,'h4/const/bool/RES_FORALL'(s(fun(A,bool),Q),s(fun(A,bool),happ(s(fun(fun(A,bool),fun(A,bool)),V__1),s(fun(A,bool),R))))))))))).
