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

Lemma lc_body_exp_wrt_exp : forall e1 e2, body_exp_wrt_exp e1 -> lc_exp e2 -> lc_exp (open_exp_wrt_exp e1 e2).