End Eq_rect_eq.
(** Export the theory of injective dependent elimination *)
Module EqdepTheory := EqdepTheory(Eq_rect_eq).
Export EqdepTheory.
Scheme eq_indd := Induction for eq Sort Prop.
(** We derive the irrelevance of the membership property for subsets *)

(* Specifications: *)
Lemma subset_eq_compat :
forall (U:Type) (P:U->Prop) (x y:U) (p:P x) (q:P y),
x = y -> exist P x p = exist P y q.
Lemma subsetT_eq_compat :
forall (U:Type) (P:U->Prop) (x y:U) (p:P x) (q:P y),
x = y -> existT P x p = existT P y q.