Consider the following premises. forall x: S(x) <-> S_1(x). forall x: ~W(x). exists x: S(x) -> W(x). forall x: W(x) -> S(x). Can we infer forall x: S_1(x) from them?
 From exists x: S(x) -> W(x), forall x: W(x) -> S(x) we can infer exists x: S(x) <-> W(x). From exists x: S(x) <-> W(x), forall x: ~W(x) we can infer exists x: ~S(x). Finally, from forall x: S(x) <-> S_1(x), exists x: ~S(x) we can infer exists x: ~S_1(x), which contradicts forall x: S_1(x). Therefore, the answer is no.