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