Consider the following premises. forall y_4: P(y_4) <-> S(y_4). forall y_4: P(y_4) or ~S(y_4). Can we infer exists y_4: ~(P(y_4) <-> S(y_4)) from them?
 From forall y_4: P(y_4) or ~S(y_4) we can infer forall y_4: S(y_4) -> P(y_4). From forall y_4: P(y_4) <-> S(y_4) we can infer forall y_4: P(y_4) -> S(y_4). Finally, from forall y_4: P(y_4) -> S(y_4), forall y_4: S(y_4) -> P(y_4) we can infer forall y_4: P(y_4) <-> S(y_4), which contradicts exists y_4: ~(P(y_4) <-> S(y_4)). Therefore, the answer is no.