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 holds
( J,v |= All (x,p) iff 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 holds
( J,v |= All (x,p) iff 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 holds
( J,v |= All (x,p) iff 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 holds
( J,v |= All (x,p) iff 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 |= All (x,p) iff 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 )

A1: now

assume A2: 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: J,v |= All (x,p)
for w being Element of Valuations_in A st ( for y being bound_QC-variable st x <> y holds
w . y = v . y ) holds
(Valid (p,J)) . w = TRUE

proof

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

end;

hence J,v |= All (x,p) by Th31; :: thesis: verum

end;

now

assume A3: J,v |= All (x,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
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 for y being bound_QC-variable st x <> y holds
w . y = v . y ; :: thesis: J,w |= p
then (Valid (p,J)) . w = TRUE by A3, Th31;
hence J,w |= p by Def12; :: thesis: verum

end;

hence ( J,v |= All (x,p) iff 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 ) by A1; :: thesis: verum