End Dependent_Equality.
Arguments eq_dep  [U P] p x q _.
Arguments eq_dep1 [U P] p x q y.
(** Dependent equality is equivalent to equality on dependent pairs *)

(* Specifications: *)
Lemma eq_sigT_eq_dep :
forall (U:Type) (P:U -> Type) (p q:U) (x:P p) (y:P q),
existT P p x = existT P q y -> eq_dep p x q y.
Lemma eq_dep_eq_sigT :
forall (U:Type) (P:U -> Type) (p q:U) (x:P p) (y:P q),
eq_dep p x q y -> existT P p x = existT P q y.
Lemma eq_sigT_iff_eq_dep :
forall (U:Type) (P:U -> Type) (p q:U) (x:P p) (y:P q),
existT P p x = existT P q y <-> eq_dep p x q y.