%   ORIGINAL: h4/ConseqConv/exists__eq__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
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Goal: !Q P. (!s. P s <=> Q s) ==> ((?s. P s) <=> (?s. Q s))
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Goal: !Q P. (!s. happ P s <=> happ Q s) ==> ((?s. happ P s) <=> (?s. happ Q s))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q84551,TV_Q84547]: ![V_f, V_g]: (![V_x]: s(TV_Q84547,happ(s(t_fun(TV_Q84551,TV_Q84547),V_f),s(TV_Q84551,V_x))) = s(TV_Q84547,happ(s(t_fun(TV_Q84551,TV_Q84547),V_g),s(TV_Q84551,V_x))) => s(t_fun(TV_Q84551,TV_Q84547),V_f) = s(t_fun(TV_Q84551,TV_Q84547),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ch4s_ConseqConvs_existsu_u_equ_u_thm, conjecture, ![TV_u_27a]: ![V_Q, V_P]: (![V_s]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_s))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_s))) => (?[V_s]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_s)))) <=> ?[V_s]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_s))))))).
