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