Consider the following premises. forall z: W(z) -> W_2(z). forall z: W(z) <-> W_2(z). forall z: ~W_2(z). Can we infer forall z: ~W(z) from them? If possible, name the inference rules being used at each step.
 From forall z: W(z) <-> W_2(z) we can infer forall z: W_2(z) -> W(z) via universal biconditional elimination. From forall z: W(z) -> W_2(z), forall z: W_2(z) -> W(z) we can infer forall z: W(z) <-> W_2(z) via universal biconditional introduction. Finally, from forall z: W(z) <-> W_2(z), forall z: ~W_2(z) we can infer forall z: ~W(z) via universal biconditional elimination. Therefore, the answer is yes.