Section Proof_irrelevance_gen.
Variable bool : Prop.
Variable true : bool.
Variable false : bool.
Hypothesis bool_elim : forall C:Prop, C -> C -> bool -> C.
Hypothesis
bool_elim_redl : forall (C:Prop) (c1 c2:C), c1 = bool_elim C c1 c2 true.
Hypothesis
bool_elim_redr : forall (C:Prop) (c1 c2:C), c2 = bool_elim C c1 c2 false.
Let bool_dep_induction :=
forall P:bool -> Prop, P true -> P false -> forall b:bool, P b.

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