let x be bound_QC-variable; :: thesis: for A being non empty set
for J being interpretation of A
for v being Element of Valuations_in A
for S being Element of CQC-Sub-WFF
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val (v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)),S,x,xSQ) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S )
let A be non empty set ; :: thesis: for J being interpretation of A
for v being Element of Valuations_in A
for S being Element of CQC-Sub-WFF
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val (v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)),S,x,xSQ) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S )
let J be interpretation of A; :: thesis: for v being Element of Valuations_in A
for S being Element of CQC-Sub-WFF
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val (v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)),S,x,xSQ) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S )
let v be Element of Valuations_in A; :: thesis: for S being Element of CQC-Sub-WFF
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val (v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)),S,x,xSQ) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S )
let S be Element of CQC-Sub-WFF ; :: thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val (v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)),S,x,xSQ) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S )
let xSQ be second_Q_comp of [S,x]; :: thesis: ( [S,x] is quantifiable implies ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val (v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)),S,x,xSQ) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S ) )
assume A1:
[S,x] is quantifiable
; :: thesis: ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val (v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)),S,x,xSQ) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S )
set S1 = CQCSub_All [S,x],xSQ;
set z = S_Bound (@ (CQCSub_All [S,x],xSQ));
thus
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val (v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)),S,x,xSQ) +* (x | a)) |= S ) implies for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S )
:: thesis: ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S ) implies for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val (v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)),S,x,xSQ) +* (x | a)) |= S )proof
assume A2:
for
a being
Element of
A holds
J,
(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val (v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)),S,x,xSQ) +* (x | a)) |= S
;
:: thesis: for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S
let a be
Element of
A;
:: thesis: J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S
NEx_Val (v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)),
S,
x,
xSQ = NEx_Val v,
S,
x,
xSQ
by A1, Th56;
hence
J,
(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S
by A2;
:: thesis: verum
end;
thus
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S ) implies for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val (v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)),S,x,xSQ) +* (x | a)) |= S )
:: thesis: verumproof
assume A3:
for
a being
Element of
A holds
J,
(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S
;
:: thesis: for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val (v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)),S,x,xSQ) +* (x | a)) |= S
let a be
Element of
A;
:: thesis: J,(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val (v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)),S,x,xSQ) +* (x | a)) |= S
NEx_Val (v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)),
S,
x,
xSQ = NEx_Val v,
S,
x,
xSQ
by A1, Th56;
hence
J,
(v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)) . ((NEx_Val (v . ((S_Bound (@ (CQCSub_All [S,x],xSQ))) | a)),S,x,xSQ) +* (x | a)) |= S
by A3;
:: thesis: verum
end;