Section Axioms.
Variable U:Type.
(** Invariance by Substitution of Reflexive Equality Proofs *)
Proof M.eq_rect_eq U.
Proof (fun p Q => M.eq_rect_eq U p Q).
(** Injectivity of Dependent Equality *)
Proof (eq_rect_eq__eq_dep_eq U eq_rect_eq).
(** Uniqueness of Identity Proofs (UIP) is a consequence of *)
(** Injectivity of Dependent Equality *)
Proof (eq_dep_eq__UIP U eq_dep_eq).
(** Uniqueness of Reflexive Identity Proofs is a direct instance of UIP *)
Proof (UIP__UIP_refl U UIP).
(** Streicher's axiom K is a direct consequence of Uniqueness of Reflexive Identity Proofs *)
Proof (UIP_refl__Streicher_K U UIP_refl).
End Axioms.
(** UIP implies the injectivity of equality on dependent pairs in Type *)

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