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 v1 being Element of Valuations_in A st ( for y being bound_QC-variable st x <> y holds
v1 . y = v . y ) holds
(Valid (p,J)) . v1 = TRUE )
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 v1 being Element of Valuations_in A st ( for y being bound_QC-variable st x <> y holds
v1 . y = v . y ) holds
(Valid (p,J)) . v1 = TRUE )
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 v1 being Element of Valuations_in A st ( for y being bound_QC-variable st x <> y holds
v1 . y = v . y ) holds
(Valid (p,J)) . v1 = TRUE )
let p be Element of CQC-WFF ; :: thesis: for J being interpretation of A holds
( J,v |= All (x,p) 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
(Valid (p,J)) . v1 = TRUE )
let J be interpretation of A; :: thesis: ( J,v |= All (x,p) 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
(Valid (p,J)) . v1 = TRUE )

hereby :: 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
(Valid (p,J)) . v1 = TRUE ) implies J,v |= All (x,p) )

assume J,v |= All (x,p) ; :: 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
(Valid (p,J)) . v1 = TRUE
then (FOR_ALL (x,(Valid (p,J)))) . v = TRUE by Th30;
hence for v1 being Element of Valuations_in A st ( for y being bound_QC-variable st x <> y holds
v1 . y = v . y ) holds
(Valid (p,J)) . v1 = TRUE by Th8; :: thesis: verum

end;

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
(Valid (p,J)) . v1 = TRUE ; :: thesis: J,v |= All (x,p)
then (FOR_ALL (x,(Valid (p,J)))) . v = TRUE by Th8;
hence J,v |= All (x,p) by Th30; :: thesis: verum