let p be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable
for f being FinSequence of CQC-WFF st Suc f = All x,p & Ant f |= Suc f holds
for y being bound_QC-variable holds Ant f |= p . x,y
let x be bound_QC-variable; :: thesis: for f being FinSequence of CQC-WFF st Suc f = All x,p & Ant f |= Suc f holds
for y being bound_QC-variable holds Ant f |= p . x,y
let f be FinSequence of CQC-WFF ; :: thesis: ( Suc f = All x,p & Ant f |= Suc f implies for y being bound_QC-variable holds Ant f |= p . x,y )
assume A1: ( Suc f = All x,p & Ant f |= Suc f ) ; :: thesis: for y being bound_QC-variable holds Ant f |= p . x,y
let y be bound_QC-variable; :: thesis: Ant f |= p . x,y
let A be non empty set ; :: according to CALCUL_1:def 15 :: thesis: for J being interpretation of A
for v being Element of Valuations_in A st J,v |= Ant f holds
J,v |= p . x,y
let J be interpretation of A; :: thesis: for v being Element of Valuations_in A st J,v |= Ant f holds
J,v |= p . x,y
let v be Element of Valuations_in A; :: thesis: ( J,v |= Ant f implies J,v |= p . x,y )
assume J,v |= Ant f ; :: thesis: J,v |= p . x,y
then A2: J,v |= All x,p by A1, Def15;
ex a being Element of A st
( v . y = a & J,v . (x | a) |= p )

proof

take a = v . y; :: thesis: ( v . y = a & J,v . (x | a) |= p )
thus ( v . y = a & J,v . (x | a) |= p ) by A2, SUBLEMMA:51; :: thesis: verum

end;

hence J,v |= p . x,y by Th24; :: thesis: verum