Consider the following premises. exists x: S(x) -> S_1(x). forall x: S(x) or ~S_1(x). Can we infer exists x: S_1(x) -> S(x) from them?
 From forall x: S(x) or ~S_1(x) we can infer forall x: S_1(x) -> S(x). From exists x: S(x) -> S_1(x), forall x: S_1(x) -> S(x) we can infer exists x: S(x) <-> S_1(x). Finally, from exists x: S(x) <-> S_1(x) we can infer exists x: S_1(x) -> S(x). Therefore, the answer is yes.