Definition bool_fun_if (x : BDDvar) (bf bf' : bool_fun) : bool_fun :=
  fun vb : var_binding => ifb (vb x) (bf vb) (bf' vb).

Lemma bool_fun_if_lemma_1 : forall (x : BDDvar) (bfl1 bfl2 bfr1 bfr2 : bool_fun), bool_fun_eq (bool_fun_if x (bool_fun_or bfr1 bfr2) (bool_fun_or bfl1 bfl2)) (bool_fun_or (bool_fun_if x bfr1 bfl1) (bool_fun_if x bfr2 bfl2)).