Consider the following premises. forall z_3: Q_4(z_3) or ~S(z_3). exists z_3: Q_4(z_3) -> S(z_3). forall z_3: Q_4(z_3) -> S(z_3). Can we infer forall z_3: ~(S(z_3) -> Q_4(z_3)) from them?
That contradicts the premises. Therefore the answer is no.