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