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').