Consider the following premises. forall z_2: Q(z_2) -> R_1(z_2). exists z_2: W(z_2) -> Q(z_2). forall z_2: R_1(z_2) <-> W(z_2). Can we infer forall z_2: ~(W(z_2) -> R_1(z_2)) from them?
That contradicts the premises. Therefore the answer is no.