Consider the following premises. exists y: U_2(y) <-> P(y). forall y: P(y) <-> U_2(y). Can we infer exists y: P(y) <-> U_2(y) from them?
 From forall y: P(y) <-> U_2(y) we can infer forall y: P(y) -> U_2(y). From exists y: U_2(y) <-> P(y) we can infer exists y: U_2(y) -> P(y). Finally, from forall y: P(y) -> U_2(y), exists y: U_2(y) -> P(y) we can infer exists y: P(y) <-> U_2(y). Therefore, the answer is yes.