reserve W for Universe,
  H for ZF-formula,
  x,y,z,X for set,
  k for Variable,
  f for Function of VAR,W,
  u,v for Element of W;

theorem Th5:
  for H st { x.0,x.1,x.2 } misses Free H holds W |=
  the_axiom_of_substitution_for H
proof
  for H,f st { x.0,x.1,x.2 } misses Free H & W,f |= All(x.3,Ex(x.0,All(x.4
  ,H <=> x.4 '=' x.0))) for u holds def_func'(H,f).:u in W
  proof
    let H,f such that
    { x.0,x.1,x.2 } misses Free H and
    W,f |= All(x.3,Ex(x.0,All(x.4,H <=> x.4 '=' x.0)));
    let u;
    card u in card W by CLASSES2:1;
    then card (def_func'(H,f).:u) in card W by CARD_1:67,ORDINAL1:12;
    hence thesis by CLASSES1:1;
  end;
  hence thesis by ZFMODEL1:15;
end;
