Consider the following premises. forall z_1: P(z_1) -> U(z_1). forall z_1: U(z_1) -> S_3(z_1). forall z_1: ~S_3(z_1). forall z_1: U(z_1) -> P(z_1). Can we infer exists z_1: P(z_1) from them?
 From forall z_1: U(z_1) -> S_3(z_1), forall z_1: ~S_3(z_1) we can infer forall z_1: ~U(z_1). From forall z_1: U(z_1) -> P(z_1), forall z_1: P(z_1) -> U(z_1) we can infer forall z_1: U(z_1) <-> P(z_1). Finally, from forall z_1: U(z_1) <-> P(z_1), forall z_1: ~U(z_1) we can infer forall z_1: ~P(z_1), which contradicts exists z_1: P(z_1). Therefore, the answer is no.