Consider the following premises. forall x: Q_1(x) -> P(x). exists x: P(x) -> S_4(x). forall x: S_4(x) -> Q_1(x). forall x: Q_1(x) -> S_4(x). Can we infer forall x: ~(S_4(x) <-> P(x)) from them?
That contradicts the premises. Therefore the answer is no.