Consider the following premises. forall z_4: S(z_4) or ~P_2(z_4). exists z_4: P_2(z_4) <-> S(z_4). Can we infer forall z_4: ~(P_2(z_4) -> S(z_4)) from them? If possible, name the inference rules being used at each step.
 From exists z_4: P_2(z_4) <-> S(z_4) we can infer exists z_4: S(z_4) -> P_2(z_4) via existential biconditional elimination. Finally, from forall z_4: S(z_4) or ~P_2(z_4) we can infer forall z_4: P_2(z_4) -> S(z_4) via universal rewriting, which contradicts forall z_4: ~(P_2(z_4) -> S(z_4)). Therefore, the answer is no.