Consider the following premises. exists x: P_1(x) <-> S(x). forall x: S(x) -> P_1(x). forall x: ~P_1(x). Can we infer exists x: ~S(x) from them?
 From exists x: P_1(x) <-> S(x) we can infer exists x: P_1(x) -> S(x). From forall x: S(x) -> P_1(x), exists x: P_1(x) -> S(x) we can infer exists x: S(x) <-> P_1(x). Finally, from exists x: S(x) <-> P_1(x), forall x: ~P_1(x) we can infer exists x: ~S(x). Therefore, the answer is yes.