Consider the following premises. exists z_1: W(z_1) <-> S(z_1). forall z_1: S(z_1) <-> S_4(z_1). forall z_1: W(z_1) -> S_4(z_1). Can we infer forall z_1: ~(W(z_1) -> S(z_1)) from them?
That contradicts the premises. Therefore the answer is no.