(* If we also assume the excluded middle, we can derive         *)
(* the usual replacement schemata.                              *)

Definition functional (P : Ens -> Ens -> Prop) :=
  forall x y y' : Ens, P x y -> P x y' -> EQ y y'.Definition replacement :=
  forall P : Ens -> Ens -> Prop,
  functional P ->
  (forall x y y' : Ens, EQ y y' -> P x y -> P x y') ->
  (forall x x' y : Ens, EQ x x' -> P x y -> P x' y) ->
  forall X : Ens,
  EXType _
    (fun Y : Ens =>
     forall y : Ens,
     (IN y Y -> EXType _ (fun x : Ens => IN x X /\ P x y)) /\
     (EXType _ (fun x : Ens => IN x X /\ P x y) -> IN y Y)).

Theorem classical_Collection_Replacement : (forall S : Prop, S \/ (S -> False)) -> collection -> replacement.