let H2, H1 be ZF-formula; :: thesis: for x being Variable
for M being non empty set st not x in Free H2 holds
M |= (All x,(H1 => H2)) => ((Ex x,H1) => H2)
let x be Variable; :: thesis: for M being non empty set st not x in Free H2 holds
M |= (All x,(H1 => H2)) => ((Ex x,H1) => H2)
let M be non empty set ; :: thesis: ( not x in Free H2 implies M |= (All x,(H1 => H2)) => ((Ex x,H1) => H2) )
assume A1: not x in Free H2 ; :: thesis: M |= (All x,(H1 => H2)) => ((Ex x,H1) => H2)
let v be Function of VAR ,M; :: according to ZF_MODEL:def 5 :: thesis: M,v |= (All x,(H1 => H2)) => ((Ex x,H1) => H2)

now

assume A2: M,v |= All x,(H1 => H2) ; :: thesis: M,v |= (Ex x,H1) => H2

now

assume M,v |= Ex x,H1 ; :: thesis: M,v |= H2
then consider m being Element of M such that
A3: M,v / x,m |= H1 by Th82;
M,v / x,m |= H1 => H2 by A2, Th80;
then M,v / x,m |= H2 by A3, ZF_MODEL:18;
then M,v / x,m |= All x,H2 by A1, ZFMODEL1:10;
then M,v |= All x,H2 by Th81;
then M,v / x,(v . x) |= H2 by Th80;
hence M,v |= H2 by FUNCT_7:37; :: thesis: verum

end;

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

end;

hence M,v |= (All x,(H1 => H2)) => ((Ex x,H1) => H2) by ZF_MODEL:18; :: thesis: verum