let Al be QC-alphabet ; :: thesis: for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for f being FinSequence of CQC-WFF Al st Suc f = All (x,p) & Ant f |= Suc f holds
for y being bound_QC-variable of Al holds Ant f |= p . (x,y)
let p be Element of CQC-WFF Al; :: thesis: for x being bound_QC-variable of Al
for f being FinSequence of CQC-WFF Al st Suc f = All (x,p) & Ant f |= Suc f holds
for y being bound_QC-variable of Al holds Ant f |= p . (x,y)
let x be bound_QC-variable of Al; :: thesis: for f being FinSequence of CQC-WFF Al st Suc f = All (x,p) & Ant f |= Suc f holds
for y being bound_QC-variable of Al holds Ant f |= p . (x,y)
let f be FinSequence of CQC-WFF Al; :: thesis: ( Suc f = All (x,p) & Ant f |= Suc f implies for y being bound_QC-variable of Al holds Ant f |= p . (x,y) )
assume A1: ( Suc f = All (x,p) & Ant f |= Suc f ) ; :: thesis: for y being bound_QC-variable of Al holds Ant f |= p . (x,y)
let y be bound_QC-variable of Al; :: thesis: Ant f |= p . (x,y)
let A be non empty set ; :: according to CALCUL_1:def 15 :: thesis: for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds
J,v |= p . (x,y)
let J be interpretation of Al,A; :: thesis: for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds
J,v |= p . (x,y)
let v be Element of Valuations_in (Al,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;
ex a being Element of A st
( v . y = a & J,v . (x | a) |= p )
proof
take v . y ; :: thesis: ( v . y = v . y & J,v . (x | (v . y)) |= p )
thus ( v . y = v . y & J,v . (x | (v . y)) |= p ) by A2, SUBLEMMA:50; :: thesis: verum
end;
hence J,v |= p . (x,y) by Th24; :: thesis: verum