let A be QC-alphabet ; :: thesis: for S being Element of QC-Sub-WFF A ex p being QC-formula of A ex e being Element of vSUB A st S = [p,e]
let S be Element of QC-Sub-WFF A; :: thesis: ex p being QC-formula of A ex e being Element of vSUB A st S = [p,e]
( [:(QC-WFF A),(vSUB A):] is Subset of [:([:NAT,(QC-symbols A):] *),(vSUB A):] & ( for k being Nat
for p being QC-pred_symbol of k,A
for ll being QC-variable_list of k,A
for e being Element of vSUB A holds [(<*p*> ^ ll),e] in [:(QC-WFF A),(vSUB A):] ) & ( for e being Element of vSUB A holds [<*[0,0]*>,e] in [:(QC-WFF A),(vSUB A):] ) & ( for p being FinSequence of [:NAT,(QC-symbols A):]
for e being Element of vSUB A st [p,e] in [:(QC-WFF A),(vSUB A):] holds
[(<*[1,0]*> ^ p),e] in [:(QC-WFF A),(vSUB A):] ) & ( for p, q being FinSequence of [:NAT,(QC-symbols A):]
for e being Element of vSUB A st [p,e] in [:(QC-WFF A),(vSUB A):] & [q,e] in [:(QC-WFF A),(vSUB A):] holds
[((<*[2,0]*> ^ p) ^ q),e] in [:(QC-WFF A),(vSUB A):] ) & ( for x being bound_QC-variable of A
for p being FinSequence of [:NAT,(QC-symbols A):]
for e being Element of vSUB A st [p,((QSub A) . [((<*[3,0]*> ^ <*x*>) ^ p),e])] in [:(QC-WFF A),(vSUB A):] holds
[((<*[3,0]*> ^ <*x*>) ^ p),e] in [:(QC-WFF A),(vSUB A):] ) ) by Th7;
then [:(QC-WFF A),(vSUB A):] is A -Sub-closed ;
then QC-Sub-WFF A c= [:(QC-WFF A),(vSUB A):] by Def17;
then S in [:(QC-WFF A),(vSUB A):] ;
then consider a, b being object such that
A1: a in QC-WFF A and
A2: b in vSUB A and
A3: S = [a,b] by ZFMISC_1:def 2;
reconsider e = b as Element of vSUB A by A2;
reconsider p = a as Element of QC-WFF A by A1;
take p ; :: thesis: ex e being Element of vSUB A st S = [p,e]
take e ; :: thesis: S = [p,e]
thus S = [p,e] by A3; :: thesis: verum