reserve F,H,H9 for ZF-formula,
  x,y,z,t for Variable,
  a,b,c,d,A,X for set;
reserve E for non empty set,
  f,g,h for Function of VAR,E,
  v1,v2,v3,v4,v5,u5 for Element of VAL E;

theorem
  E,f |= Ex(x,y,H) iff ex g st (for z st g.z <> f.z holds x = z or y = z
  ) & E,g |= H
proof
  thus E,f |= Ex(x,y,H) implies ex g st (for z st g.z <> f.z holds x = z or y
  = z) & E,g |= H
  proof
    assume E,f |= Ex(x,y,H);
    then consider g such that
A1: for z st g.z <> f.z holds x = z and
A2: E,g |= Ex(y,H) by Th20;
    consider h such that
A3: for z st h.z <> g.z holds y = z and
A4: E,h |= H by A2,Th20;
    take h;
    thus for z st h.z <> f.z holds x = z or y = z
    proof
      let z such that
A5:   h.z <> f.z and
A6:   x <> z and
A7:   y <> z;
      g.z = f.z by A1,A6;
      hence contradiction by A3,A5,A7;
    end;
    thus thesis by A4;
  end;
  given g such that
A8: for z st g.z <> f.z holds x = z or y = z and
A9: E,g |= H;
  set h = f+*(x,g.x);
  now
    let z;
    assume that
A10: g.z <> h.z and
A11: y <> z;
A12: x <> z by A10,FUNCT_7:128;
    then g.z = f.z by A8,A11;
    hence contradiction by A10,A12,FUNCT_7:32;
  end;
  then
A13: E,h |= Ex(y,H) by A9,Th20;
  for z st h.z <> f.z holds x = z by FUNCT_7:32;
  hence thesis by A13,Th20;
end;
