Consider the following premises. forall x: P(x) <-> U(x). forall x: P(x) -> U(x). Can we infer forall x: U(x) <-> P(x) from them?
 From forall x: P(x) <-> U(x) we can infer forall x: U(x) -> P(x). Finally, from forall x: U(x) -> P(x), forall x: P(x) -> U(x) we can infer forall x: U(x) <-> P(x). Therefore, the answer is yes.