let H1, H2 be ZF-formula; :: thesis: for x being Variable
for M being non empty set st M |= (Ex x,H1) => H2 holds
M |= H1 => H2
let x be Variable; :: thesis: for M being non empty set st M |= (Ex x,H1) => H2 holds
M |= H1 => H2
let M be non empty set ; :: thesis: ( M |= (Ex x,H1) => H2 implies M |= H1 => H2 )
assume A1: for v being Function of VAR ,M holds M,v |= (Ex x,H1) => H2 ; :: according to ZF_MODEL:def 5 :: thesis: M |= H1 => H2
let v be Function of VAR ,M; :: according to ZF_MODEL:def 5 :: thesis: M,v |= H1 => H2
A2: M,v |= (Ex x,H1) => H2 by A1;

now

assume M,v |= H1 ; :: thesis: M,v |= H2
then M,v / x,(v . x) |= H1 by FUNCT_7:37;
then M,v |= Ex x,H1 by Th82;
hence M,v |= H2 by A2, ZF_MODEL:18; :: thesis: verum

end;

hence M,v |= H1 => H2 by ZF_MODEL:18; :: thesis: verum