let Al be QC-alphabet ; :: thesis: for A being non empty set
for x being bound_QC-variable of Al
for v being Element of Valuations_in (Al,A)
for p being Element of CQC-WFF Al
for J being interpretation of Al,A st J,v |= p & not x in still_not-bound_in p holds
for w being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
w . y = v . y ) holds
J,w |= p
let A be non empty set ; :: thesis: for x being bound_QC-variable of Al
for v being Element of Valuations_in (Al,A)
for p being Element of CQC-WFF Al
for J being interpretation of Al,A st J,v |= p & not x in still_not-bound_in p holds
for w being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
w . y = v . y ) holds
J,w |= p
let x be bound_QC-variable of Al; :: thesis: for v being Element of Valuations_in (Al,A)
for p being Element of CQC-WFF Al
for J being interpretation of Al,A st J,v |= p & not x in still_not-bound_in p holds
for w being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
w . y = v . y ) holds
J,w |= p
let v be Element of Valuations_in (Al,A); :: thesis: for p being Element of CQC-WFF Al
for J being interpretation of Al,A st J,v |= p & not x in still_not-bound_in p holds
for w being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
w . y = v . y ) holds
J,w |= p
let p be Element of CQC-WFF Al; :: thesis: for J being interpretation of Al,A st J,v |= p & not x in still_not-bound_in p holds
for w being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
w . y = v . y ) holds
J,w |= p
let J be interpretation of Al,A; :: thesis: ( J,v |= p & not x in still_not-bound_in p implies for w being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
w . y = v . y ) holds
J,w |= p )
assume that
A1: J,v |= p and
A2: not x in still_not-bound_in p ; :: thesis: for w being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
w . y = v . y ) holds
J,w |= p

now :: thesis: for w being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
w . y = v . y ) holds
J,w |= p

let w be Element of Valuations_in (Al,A); :: thesis: ( ( for y being bound_QC-variable of Al st x <> y holds
w . y = v . y ) implies J,w |= p )
assume A3: for y being bound_QC-variable of Al st x <> y holds
w . y = v . y ; :: thesis: J,w |= p
(Valid (p,J)) . v = TRUE by A1, Def7;
then (Valid (p,J)) . w = TRUE by A2, A3, Th27;
hence J,w |= p by Def7; :: thesis: verum

end;

hence for w being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
w . y = v . y ) holds
J,w |= p ; :: thesis: verum