Consider the following premises. forall z: W(z) -> P_2(z). forall z: P(z) -> W(z). exists z: P(z) or ~P_2(z). Can we infer exists z: P_2(z) <-> P(z) from them? If possible, name the inference rules being used at each step.
 From exists z: P(z) or ~P_2(z) we can infer exists z: P_2(z) -> P(z) via existential rewriting. From forall z: P(z) -> W(z), forall z: W(z) -> P_2(z) we can infer forall z: P(z) -> P_2(z) via universal transitivity. Finally, from exists z: P_2(z) -> P(z), forall z: P(z) -> P_2(z) we can infer exists z: P_2(z) <-> P(z) via existential biconditional introduction. Therefore, the answer is yes.