Proof with auto.
intros.
unfold open_ee.
apply subst_ee_open_ee_rec...
Qed.

(* Specifications: *)
Lemma subst_ee_open_ee_var : forall (x y:atom) u e,
y <> x ->
expr u ->
open_ee (subst_ee x u e) y = subst_ee x u (open_ee e y).