Definition bool_fun_iff (bf1 bf2 : bool_fun) : bool_fun :=
  fun vb : var_binding => eqb (bf1 vb) (bf2 vb).

Lemma bool_fun_iff_is_and_impl_impl : forall bf1 bf2 : bool_fun, bool_fun_eq (bool_fun_iff bf1 bf2) (bool_fun_and (bool_fun_impl bf1 bf2) (bool_fun_impl bf2 bf1)).