Module N:=DecidableEqDep(M).
(** Invariance by Substitution of Reflexive Equality Proofs *)
Proof eq_rect_eq_dec eq_dec.
(** Injectivity of Dependent Equality *)
Proof (eq_rect_eq__eq_dep_eq U eq_rect_eq).
(** Uniqueness of Identity Proofs (UIP) *)
Proof (eq_dep_eq__UIP U eq_dep_eq).
(** Uniqueness of Reflexive Identity Proofs *)
Proof (UIP__UIP_refl U UIP).
(** Streicher's axiom K *)
Proof (K_dec_type eq_dec).
(** Proof-irrelevance on subsets of decidable sets *)
Proof N.inj_pairP2.
(** Injectivity of equality on dependent pairs in [Type] *)

Lemma inj_pair2 :
forall (P:U -> Type) (p:U) (x y:P p),
existT P p x = existT P p y -> x = y.