(** *** Exercise [subst_open_var] *)
(** The lemma above is most often used with [k = 0] and [e2] as some fresh variable. Therefore, it simplifies matters to define the following useful corollary.  HINT: Do not use induction. Rewrite with [subst_open_rec] and [subst_neq_var].  *)

Lemma subst_open_var : forall (x y : atom) u e,
y <> x ->
lc u ->
open ([x ~> u] e) y = [x ~> u] (open e y).