Consider the following premises. exists x_4: Q_4(x_4) -> P_1(x_4). forall x_4: S(x_4) -> Q_4(x_4). forall x_4: P_1(x_4) -> S(x_4). Can we infer forall x_4: ~(S(x_4) -> P_1(x_4)) from them?
That contradicts the premises. Therefore the answer is no.