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

proof

assume M,v |= Ex x,H ; :: thesis: ex m being Element of M st M,v / x,m |= H
then consider v9 being Function of VAR ,M such that
A1: ( ( for y being Variable st v9 . y <> v . y holds
x = y ) & M,v9 |= H ) by ZF_MODEL:20;
take v9 . x ; :: thesis: M,v / x,(v9 . x) |= H
thus M,v / x,(v9 . x) |= H by A1, FUNCT_7:131; :: thesis: verum

end;

given m being Element of M such that A2: M,v / x,m |= H ; :: thesis: M,v |= Ex x,H

now

take v9 = v / x,m; :: thesis: ( ( for y being Variable st v9 . y <> v . y holds
x = y ) & M,v9 |= H )
thus for y being Variable st v9 . y <> v . y holds
x = y by FUNCT_7:34; :: thesis: M,v9 |= H
thus M,v9 |= H by A2; :: thesis: verum

end;

hence M,v |= Ex x,H by ZF_MODEL:20; :: thesis: verum