Proof eq_dep_eq__inj_pair2 U N.eq_dep_eq.
(** Injectivity of equality on dependent pairs with second component in [Type] *)
Notation inj_pairT2 := inj_pair2.
End DecidableEqDepSet.

(** From decidability to inj_pair2 **)
Lemma inj_pair2_eq_dec : forall A:Type, (forall x y:A, {x=y}+{x<>y}) ->
( forall (P:A -> Type) (p:A) (x y:P p), existT P p x = existT P p y -> x = y ).