Consider the following premises. forall y: ~U(y). forall y: U(y) <-> P_2(y). exists y: U(y) -> P_2(y). Can we infer exists y: ~P_2(y) from them?
 From forall y: U(y) <-> P_2(y) we can infer forall y: P_2(y) -> U(y). From forall y: P_2(y) -> U(y), exists y: U(y) -> P_2(y) we can infer exists y: P_2(y) <-> U(y). Finally, from exists y: P_2(y) <-> U(y), forall y: ~U(y) we can infer exists y: ~P_2(y). Therefore, the answer is yes.