Consider the following premises. forall x: U(x) or ~P_1(x). exists x: P_1(x) <-> U(x). Can we infer forall x: ~(P_1(x) -> U(x)) from them?
 From exists x: P_1(x) <-> U(x) we can infer exists x: U(x) -> P_1(x). Finally, from forall x: U(x) or ~P_1(x) we can infer forall x: P_1(x) -> U(x), which contradicts forall x: ~(P_1(x) -> U(x)). Therefore, the answer is no.