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

Lemma subst_var : forall (x y : var) u e,
y <> x ->
lc_exp u ->
([x ~> u] e) ^ y = [x ~> u] (e ^ y).