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