Consider the following premises. forall z: ~U(z). forall z: U(z) <-> P_4(z). exists z: U(z) -> P_4(z). Can we infer exists z: ~P_4(z) from them?
 From forall z: U(z) <-> P_4(z) we can infer forall z: P_4(z) -> U(z). From exists z: U(z) -> P_4(z), forall z: P_4(z) -> U(z) we can infer exists z: U(z) <-> P_4(z). Finally, from exists z: U(z) <-> P_4(z), forall z: ~U(z) we can infer exists z: ~P_4(z). Therefore, the answer is yes.