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

now

let w be Element of Valuations_in A; :: thesis: ( ( for y being bound_QC-variable st x <> y holds
w . y = v . y ) implies J,w |= p )
assume A3: for y being bound_QC-variable st x <> y holds
w . y = v . y ; :: thesis: J,w |= p
(Valid p,J) . v = TRUE by A1, Def12;
then (Valid p,J) . w = TRUE by A2, A3, Th39;
hence J,w |= p by Def12; :: thesis: verum

end;

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