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 Th130:
  not x in Free H2 implies M |= All(x,H1 => H2) => (Ex(x,H1) => H2)
proof
  assume
A1: not x in Free H2;
  let v;
  now
    assume
A2: M,v |= All(x,H1 => H2);
    now
      assume M,v |= Ex(x,H1);
      then consider m such that
A3:   M,v/(x,m) |= H1 by Th73;
      M,v/(x,m) |= H1 => H2 by A2,Th71;
      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 Th72;
      then M,v/(x,v.x) |= H2 by Th71;
      hence M,v |= H2 by FUNCT_7:35;
    end;
    hence M,v |= Ex(x,H1) => H2 by ZF_MODEL:18;
  end;
  hence thesis by ZF_MODEL:18;
end;
