Consider the following premises. forall z: P(z) <-> W_3(z). forall z: P(z) or ~W_3(z). Can we infer exists z: ~(P(z) <-> W_3(z)) from them? If possible, name the inference rules being used at each step.
 From forall z: P(z) <-> W_3(z) we can infer forall z: P(z) -> W_3(z) via universal biconditional elimination. From forall z: P(z) or ~W_3(z) we can infer forall z: W_3(z) -> P(z) via universal rewriting. Finally, from forall z: P(z) -> W_3(z), forall z: W_3(z) -> P(z) we can infer forall z: P(z) <-> W_3(z) via universal biconditional introduction, which contradicts exists z: ~(P(z) <-> W_3(z)). Therefore, the answer is no.