(** Remark: [prop_extensionality] can be replaced in lemma [ext_prop_fixpoint]
    by the weakest property [provable_prop_extensionality].
*)

(************************************************************************)
(** ** CC |- prop_ext /\ dep elim on bool -> proof-irrelevance  *)

(** [proof_irrelevance] asserts equality of all proofs of a given formula *)
Definition proof_irrelevance := forall (A:Prop) (a1 a2:A), a1 = a2.

Lemma aux : prop_extensionality -> bool_dep_induction -> true = false.