Consider the following premises. exists z_1: ~P_3(z_1). forall z_1: W(z_1) -> P_3(z_1). forall z_1: P_3(z_1) -> W(z_1). Can we infer exists z_1: ~W(z_1) from them? If possible, name the inference rules being used at each step.
 From forall z_1: P_3(z_1) -> W(z_1), forall z_1: W(z_1) -> P_3(z_1) we can infer forall z_1: P_3(z_1) <-> W(z_1) via universal biconditional introduction. Finally, from forall z_1: P_3(z_1) <-> W(z_1), exists z_1: ~P_3(z_1) we can infer exists z_1: ~W(z_1) via existential biconditional elimination. Therefore, the answer is yes.