reserve p,p1,p2,q,r,F,G,G1,G2,H,H1,H2 for ZF-formula,
  x,x1,x2,y,y1,y2,z,z1,z2,s,t for Variable,
  a,X for set;
reserve M for non empty set,
  m,m9 for Element of M,
  v,v9 for Function of VAR,M;
reserve i,j for Element of NAT;

theorem
  not x in Free H2 & M |= H1 => H2 implies M |= Ex(x,H1) => H2
proof
  assume that
A1: not x in Free H2 and
A2: for v holds M,v |= H1 => H2;
  let v;
  M |= All(x,H1 => H2) => (Ex(x,H1) => H2) by A1,Th130;
  then
A3: M,v |= All(x,H1 => H2) => (Ex(x,H1) => H2);
  for m holds M,v/(x,m) |= H1 => H2 by A2;
  then M,v |= All(x,H1 => H2) by Th71;
  hence thesis by A3,ZF_MODEL:18;
end;
