Consider the following premises. forall z: P(z) -> W(z). forall z: ~S_3(z). forall z: U(z) <-> S_3(z). forall z: W(z) -> U(z). Can we infer exists z: P(z) from them?
 From forall z: P(z) -> W(z), forall z: W(z) -> U(z) we can infer forall z: P(z) -> U(z). From forall z: U(z) <-> S_3(z), forall z: ~S_3(z) we can infer forall z: ~U(z). Finally, from forall z: P(z) -> U(z), forall z: ~U(z) we can infer forall z: ~P(z), which contradicts exists z: P(z). Therefore, the answer is no.