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 Funcs ((Valuations_in (Al,A)),BOOLEAN) holds
( (FOR_ALL (x,p)) . v = TRUE iff for v1 being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
v1 . y = v . y ) holds
p . v1 = TRUE )
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 Funcs ((Valuations_in (Al,A)),BOOLEAN) holds
( (FOR_ALL (x,p)) . v = TRUE iff for v1 being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
v1 . y = v . y ) holds
p . v1 = TRUE )
let x be bound_QC-variable of Al; :: thesis: for v being Element of Valuations_in (Al,A)
for p being Element of Funcs ((Valuations_in (Al,A)),BOOLEAN) holds
( (FOR_ALL (x,p)) . v = TRUE iff for v1 being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
v1 . y = v . y ) holds
p . v1 = TRUE )
let v be Element of Valuations_in (Al,A); :: thesis: for p being Element of Funcs ((Valuations_in (Al,A)),BOOLEAN) holds
( (FOR_ALL (x,p)) . v = TRUE iff for v1 being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
v1 . y = v . y ) holds
p . v1 = TRUE )
let p be Element of Funcs ((Valuations_in (Al,A)),BOOLEAN); :: thesis: ( (FOR_ALL (x,p)) . v = TRUE iff for v1 being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
v1 . y = v . y ) holds
p . v1 = TRUE )

A1: now :: thesis: ( (FOR_ALL (x,p)) . v = TRUE implies for v1 being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
v1 . y = v . y ) holds
p . v1 = TRUE )

assume (FOR_ALL (x,p)) . v = TRUE ; :: thesis: for v1 being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
v1 . y = v . y ) holds
p . v1 = TRUE
then ALL { (p . v99) where v99 is Element of Valuations_in (Al,A) : for y being bound_QC-variable of Al st x <> y holds
v99 . y = v . y } = TRUE by Def2;
then A2: not FALSE in { (p . v99) where v99 is Element of Valuations_in (Al,A) : for y being bound_QC-variable of Al st x <> y holds
v99 . y = v . y } by MARGREL1:17;
thus for v1 being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
v1 . y = v . y ) holds
p . v1 = TRUE :: thesis: verum

proof

let v1 be Element of Valuations_in (Al,A); :: thesis: ( ( for y being bound_QC-variable of Al st x <> y holds
v1 . y = v . y ) implies p . v1 = TRUE )
assume for y being bound_QC-variable of Al st x <> y holds
v1 . y = v . y ; :: thesis: p . v1 = TRUE
then not p . v1 = FALSE by A2;
hence p . v1 = TRUE by XBOOLEAN:def 3; :: thesis: verum

end;

end;

now :: thesis: ( ( for v1 being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
v1 . y = v . y ) holds
p . v1 = TRUE ) implies (FOR_ALL (x,p)) . v = TRUE )

assume for v1 being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
v1 . y = v . y ) holds
p . v1 = TRUE ; :: thesis: (FOR_ALL (x,p)) . v = TRUE
then for v1 being Element of Valuations_in (Al,A) holds
( not p . v1 = FALSE or ex y being bound_QC-variable of Al st
( x <> y & not v1 . y = v . y ) ) ;
then not FALSE in { (p . v99) where v99 is Element of Valuations_in (Al,A) : for y being bound_QC-variable of Al st x <> y holds
v99 . y = v . y } ;
then ALL { (p . v99) where v99 is Element of Valuations_in (Al,A) : for y being bound_QC-variable of Al st x <> y holds
v99 . y = v . y } = TRUE by MARGREL1:17;
hence (FOR_ALL (x,p)) . v = TRUE by Def2; :: thesis: verum

end;

hence ( (FOR_ALL (x,p)) . v = TRUE iff for v1 being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
v1 . y = v . y ) holds
p . v1 = TRUE ) by A1; :: thesis: verum