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_preserves_eq : forall (x : BDDvar) (bf1 bf2 bf1' bf2' : bool_fun), bool_fun_eq bf1 bf1' -> bool_fun_eq bf2 bf2' -> bool_fun_eq (bool_fun_if x bf1 bf2) (bool_fun_if x bf1' bf2').