Consider the following premises. exists y: P_3(y) -> U(y). forall y: U(y) -> P_3(y). forall y: U(y). Can we infer exists y: P_3(y) from them?
 From exists y: P_3(y) -> U(y), forall y: U(y) -> P_3(y) we can infer exists y: P_3(y) <-> U(y). From exists y: P_3(y) <-> U(y) we can infer exists y: U(y) -> P_3(y). Finally, from exists y: U(y) -> P_3(y), forall y: U(y) we can infer exists y: P_3(y). Therefore, the answer is yes.