Hint Extern 1 (lc_exp (abs _)) =>
  let x1 := fresh in
  pick_fresh x1;
  apply (lc_abs_exists x1).

Lemma subst_exp_open_exp_wrt_exp_var : forall e2 e1 x1 x2, x1 <> x2 -> lc_exp e1 -> open_exp_wrt_exp (subst_exp e1 x1 e2) (var_f x2) = subst_exp e1 x1 (open_exp_wrt_exp e2 (var_f x2)).