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