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

Lemma bool_fun_impl_preserves_eq : forall bf1 bf2 bf1' bf2' : bool_fun, bool_fun_eq bf1 bf1' -> bool_fun_eq bf2 bf2' -> bool_fun_eq (bool_fun_impl bf1 bf2) (bool_fun_impl bf1' bf2').