let Al be QC-alphabet ; :: thesis: for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
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,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
let x be bound_QC-variable of Al; :: thesis: for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
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,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
let A be non empty set ; :: thesis: for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
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,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
let J be interpretation of Al,A; :: thesis: for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
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,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
let v be Element of Valuations_in (Al,A); :: thesis: for S being Element of CQC-Sub-WFF Al
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,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
let S be Element of CQC-Sub-WFF Al; :: 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,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((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,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((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)));
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,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
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 . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) :: thesis: ( ( for a being Element of A holds J,v . ((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,x,xSQ)) +* (x | a)) |= S )

proof

set X = still_not-bound_in (S `1);
assume A2: 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 . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S
let a be Element of A; :: thesis: J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S
set V1 = (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a));
set V2 = v . ((NEx_Val (v,S,x,xSQ)) +* (x | a));
((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) by A1, Th69;
then A3: ( J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S `1 iff J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S `1 ) by Th68;
J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S by A2;
hence J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S by A3; :: thesis: verum

end;

thus ( ( for a being Element of A holds J,v . ((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,x,xSQ)) +* (x | a)) |= S ) :: thesis: verum

proof

set X = still_not-bound_in (S `1);
assume A4: for a being Element of A holds J,v . ((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
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
set V1 = (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a));
set V2 = v . ((NEx_Val (v,S,x,xSQ)) +* (x | a));
((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) by A1, Th69;
then A5: ( J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S `1 iff J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S `1 ) by Th68;
J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S by A4;
hence J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S by A5; :: thesis: verum

end;