Consider the following premises. forall y_4: P(y_4) -> S(y_4). forall y_4: Q(y_4) -> P(y_4). exists y_4: S(y_4) -> Q(y_4). Can we infer exists y_4: S(y_4) <-> P(y_4) from them?
 From exists y_4: S(y_4) -> Q(y_4), forall y_4: Q(y_4) -> P(y_4) we can infer exists y_4: S(y_4) -> P(y_4). Finally, from exists y_4: S(y_4) -> P(y_4), forall y_4: P(y_4) -> S(y_4) we can infer exists y_4: S(y_4) <-> P(y_4). Therefore, the answer is yes.