Consider the following premises. forall x: ~W(x). forall x: S_2(x) <-> P_1(x). forall x: S_2(x) <-> W(x). forall x: P_3(x) -> P_1(x). Can we infer exists x: P_3(x) from them?
 From forall x: S_2(x) <-> W(x), forall x: ~W(x) we can infer forall x: ~S_2(x). From forall x: S_2(x) <-> P_1(x), forall x: ~S_2(x) we can infer forall x: ~P_1(x). Finally, from forall x: P_3(x) -> P_1(x), forall x: ~P_1(x) we can infer forall x: ~P_3(x), which contradicts exists x: P_3(x). Therefore, the answer is no.