Consider the following premises. forall z_3: P_3(z_3) -> P(z_3). forall z_3: P_3(z_3) or ~P(z_3). exists z_3: ~P(z_3). forall z_3: P(z_3) -> P_3(z_3). Can we infer exists z_3: ~P_3(z_3) from them? If possible, name the inference rules being used at each step.
 From forall z_3: P_3(z_3) or ~P(z_3) we can infer forall z_3: P(z_3) -> P_3(z_3) via universal rewriting. From forall z_3: P(z_3) -> P_3(z_3), forall z_3: P_3(z_3) -> P(z_3) we can infer forall z_3: P(z_3) <-> P_3(z_3) via universal biconditional introduction. From forall z_3: P(z_3) <-> P_3(z_3) we can infer forall z_3: P_3(z_3) -> P(z_3) via universal biconditional elimination. From forall z_3: P(z_3) -> P_3(z_3), forall z_3: P_3(z_3) -> P(z_3) we can infer forall z_3: P(z_3) <-> P_3(z_3) via universal biconditional introduction. Finally, from forall z_3: P(z_3) <-> P_3(z_3), exists z_3: ~P(z_3) we can infer exists z_3: ~P_3(z_3) via existential biconditional elimination. Therefore, the answer is yes.