let H be ZF-formula; :: thesis: for x being Variable
for M being non empty set
for m being Element of M
for v being Function of VAR ,M holds
( M,v |= Ex x,H iff M,v / x,m |= Ex x,H )
let x be Variable; :: thesis: for M being non empty set
for m being Element of M
for v being Function of VAR ,M holds
( M,v |= Ex x,H iff M,v / x,m |= Ex x,H )
let M be non empty set ; :: thesis: for m being Element of M
for v being Function of VAR ,M holds
( M,v |= Ex x,H iff M,v / x,m |= Ex x,H )
let m be Element of M; :: thesis: for v being Function of VAR ,M holds
( M,v |= Ex x,H iff M,v / x,m |= Ex x,H )
let v be Function of VAR ,M; :: thesis: ( M,v |= Ex x,H iff M,v / x,m |= Ex x,H )
A1: for v being Function of VAR ,M
for m being Element of M st M,v |= Ex x,H holds
M,v / x,m |= Ex x,H

proof

let v be Function of VAR ,M; :: thesis: for m being Element of M st M,v |= Ex x,H holds
M,v / x,m |= Ex x,H
let m be Element of M; :: thesis: ( M,v |= Ex x,H implies M,v / x,m |= Ex x,H )
assume M,v |= Ex x,H ; :: thesis: M,v / x,m |= Ex x,H
then consider m9 being Element of M such that
A2: M,v / x,m9 |= H by Th82;
(v / x,m) / x,m9 = v / x,m9 by FUNCT_7:36;
hence M,v / x,m |= Ex x,H by A2, Th82; :: thesis: verum

end;

(v / x,m) / x,(v . x) = v / x,(v . x) by FUNCT_7:36
.= v by FUNCT_7:37 ;
hence ( M,v |= Ex x,H iff M,v / x,m |= Ex x,H ) by A1; :: thesis: verum