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

A1: now

assume (FOR_ALL x,p) . v = TRUE ; :: thesis: for v1 being Element of Valuations_in A st ( for y being bound_QC-variable st x <> y holds
v1 . y = v . y ) holds
p . v1 = TRUE
then ALL { (p . v'') where v'' is Element of Valuations_in A : for y being bound_QC-variable st x <> y holds
v'' . y = v . y } = TRUE by Def7;
then A2: not FALSE in { (p . v'') where v'' is Element of Valuations_in A : for y being bound_QC-variable st x <> y holds
v'' . y = v . y } by MARGREL1:53;
thus for v1 being Element of Valuations_in A st ( for y being bound_QC-variable st x <> y holds
v1 . y = v . y ) holds
p . v1 = TRUE :: thesis: verum

proof

let v1 be Element of Valuations_in A; :: thesis: ( ( for y being bound_QC-variable st x <> y holds
v1 . y = v . y ) implies p . v1 = TRUE )
assume for y being bound_QC-variable 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

assume for v1 being Element of Valuations_in A st ( for y being bound_QC-variable 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 A holds
( not p . v1 = FALSE or ex y being bound_QC-variable st
( x <> y & not v1 . y = v . y ) ) ;
then not FALSE in { (p . v'') where v'' is Element of Valuations_in A : for y being bound_QC-variable st x <> y holds
v'' . y = v . y } ;
then ALL { (p . v'') where v'' is Element of Valuations_in A : for y being bound_QC-variable st x <> y holds
v'' . y = v . y } = TRUE by MARGREL1:53;
hence (FOR_ALL x,p) . v = TRUE by Def7; :: thesis: verum

end;

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