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