Section big_extensionality_proof.
Variable P : Ens -> Prop.
Variable X : Ens.
Variable Y : Ens.
Hypothesis m : P X.
Hypothesis k : EQ X Y.
Let D := Doubleton X Y.
Let iXD : IN X D := Doubleton_th1 X Y.
Let iYD : IN Y D := Doubleton_th2 X Y.
Let eYX : EQ Y X := EQ_symm X Y k.
Let S := Set_Of D P.
Let iXS : IN X S := Set_Of_th1 D P X iXD m.
Let iYS : IN Y S := Extensionality Y X S eYX iXS.
Definition big_extensionality_Out : P Y := Set_Of_th2 D P Y iYS.
End big_extensionality_proof.
Proof big_extensionality_Out.
(******* here is perhaps a useful rewrite *****)

Theorem Substitute :
forall (X Y : Ens) (P : Ens -> Prop) (e : EQ X Y) (p : P Y), P X.