reserve x,y,z for Variable,
  H for ZF-formula,
  E for non empty set,
  a,b,c,X,Y,Z for set,
  u,v,w for Element of E,
  f,g,h,i,j for Function of VAR,E;

theorem
  E is epsilon-transitive implies ((for H st { x.0,x.1,x.2 } misses Free
  H holds E |= the_axiom_of_substitution_for H) iff for F being Function st F
  is_parametrically_definable_in E for X st X in E holds F.:X in E )
proof
  assume
A1: E is epsilon-transitive;
  thus (for H st { x.0,x.1,x.2 } misses Free H holds E |=
  the_axiom_of_substitution_for H) implies for F being Function st F
  is_parametrically_definable_in E for X st X in E holds F.:X in E
  by A1,Th15;
  assume
A2: for F being Function st F is_parametrically_definable_in E for X st
  X in E holds F.:X in E;
  for H,f st { x.0,x.1,x.2 } misses Free H & E,f |= All(x.3,Ex(x.0,All(x.
  4,H <=> x.4 '=' x.0))) for u holds def_func'(H,f).:u in E
  by Def4,A2;
  hence thesis by A1,Th15;
end;
