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 |= All(x,y,H) iff for g st for z st g.z <> f.z holds x = z or y =
  z holds E,g |= H
proof
  thus E,f |= All(x,y,H) implies for g st for z st g.z <> f.z holds x = z or y
  = z holds E,g |= H
  proof
    assume
A1: E,f |= All(x,y,H);
A2: for g st for z st g.z <> f.z holds x = z for h st for z st h.z <> g.z
    holds y = z holds E,h |= H
    proof
      let g;
      assume for z st g.z <> f.z holds x = z;
      then E,g |= All(y,H) by A1,Th16;
      hence thesis by Th16;
    end;
    let g such that
A3: for z st g.z <> f.z holds x = z or y = z;
    set h = g+*(y,f.y);
    for z st h.z <> f.z holds x = z
    proof
      let z such that
A4:   h.z <> f.z and
A5:   x <> z;
A6:   y <> z by A4,FUNCT_7:128;
      then g.z = f.z by A3,A5;
      hence contradiction by A4,A6,FUNCT_7:32;
    end;
    then (for z st g.z <> h.z holds y = z) implies E,g |= H by A2;
    hence thesis by FUNCT_7:32;
  end;
  assume
A7: for g st for z st g.z <> f.z holds x = z or y = z holds E,g |= H;
  now
    let g such that
A8: for z st g.z <> f.z holds x = z;
    now
      let h such that
A9:   for z st h.z <> g.z holds y = z;
      now
        let z;
        assume that
A10:    h.z <> f.z & x <> z and
A11:    y <> z;
        h.z = g.z by A9,A11;
        hence contradiction by A8,A10;
      end;
      hence E,h |= H by A7;
    end;
    hence E,g |= All(y,H) by Th16;
  end;
  hence thesis by Th16;
end;
