Section Proof_Irrelevance.
Scheme eq_dep_elim := Induction for eq Sort Type.
Definition eqbool_dep P (h1 : P true) b :=
match b return P b -> Prop with
| true => fun (h2 : P true) => h1 = h2
| false => fun (h2 : P false) => False
end.

Lemma eqbool_irrelevance : forall (b : bool) (h1 h2 : b = true), h1 = h2.