Consider the following premises. forall z_4: ~P(z_4). exists z_4: U(z_4) <-> P(z_4). forall z_4: U(z_4) -> P(z_4). Can we infer forall z_4: U(z_4) from them?
 From exists z_4: U(z_4) <-> P(z_4) we can infer exists z_4: P(z_4) -> U(z_4). From exists z_4: P(z_4) -> U(z_4), forall z_4: U(z_4) -> P(z_4) we can infer exists z_4: P(z_4) <-> U(z_4). Finally, from exists z_4: P(z_4) <-> U(z_4), forall z_4: ~P(z_4) we can infer exists z_4: ~U(z_4), which contradicts forall z_4: U(z_4). Therefore, the answer is no.