let E be non empty set ; :: thesis: for f being Function of VAR,E
for H being ZF-formula
for x being Variable holds
( E,f |= All (x,H) iff for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
E,g |= H )
let f be Function of VAR,E; :: thesis: for H being ZF-formula
for x being Variable holds
( E,f |= All (x,H) iff for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
E,g |= H )
let H be ZF-formula; :: thesis: for x being Variable holds
( E,f |= All (x,H) iff for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
E,g |= H )
let x be Variable; :: thesis: ( E,f |= All (x,H) iff for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
E,g |= H )
A1: ( ( for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
E,g |= H ) implies E,f |= H )

proof

A2: for y being Variable st f . y <> f . y holds
x = y ;
assume for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
E,g |= H ; :: thesis: E,f |= H
hence E,f |= H by A2; :: thesis: verum

end;

A3: ( E,f |= All (x,H) iff f in St ((All (x,H)),E) ) by Def4;
A4: ( ( for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
E,g |= H ) implies for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
g in St (H,E) )

proof

assume A5: for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
E,g |= H ; :: thesis: for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
g in St (H,E)
let g be Function of VAR,E; :: thesis: ( ( for y being Variable st g . y <> f . y holds
x = y ) implies g in St (H,E) )
assume for y being Variable st g . y <> f . y holds
x = y ; :: thesis: g in St (H,E)
then E,g |= H by A5;
hence g in St (H,E) by Def4; :: thesis: verum

end;

A6: ( ( for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
g in St (H,E) ) implies for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
E,g |= H )

proof

assume A7: for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
g in St (H,E) ; :: thesis: for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
E,g |= H
let g be Function of VAR,E; :: thesis: ( ( for y being Variable st g . y <> f . y holds
x = y ) implies E,g |= H )
assume for y being Variable st g . y <> f . y holds
x = y ; :: thesis: E,g |= H
then g in St (H,E) by A7;
hence E,g |= H by Def4; :: thesis: verum

end;

( E,f |= H iff f in St (H,E) ) by Def4;
hence ( E,f |= All (x,H) iff for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
E,g |= H ) by A3, A6, A4, A1, Th6; :: thesis: verum