Consider the following premises. forall z: P(z) -> U_2(z). forall z: W(z) -> P(z). forall z: U_2(z) -> W(z). forall z: W(z) -> U_2(z). Can we infer exists z: ~(W(z) <-> P(z)) from them? If possible, name the inference rules being used at each step.
 From forall z: U_2(z) -> W(z), forall z: W(z) -> U_2(z) we can infer forall z: U_2(z) <-> W(z) via universal biconditional introduction. From forall z: U_2(z) <-> W(z) we can infer forall z: U_2(z) -> W(z) via universal biconditional elimination. From forall z: P(z) -> U_2(z), forall z: U_2(z) -> W(z) we can infer forall z: P(z) -> W(z) via universal transitivity. Finally, from forall z: W(z) -> P(z), forall z: P(z) -> W(z) we can infer forall z: W(z) <-> P(z) via universal biconditional introduction, which contradicts exists z: ~(W(z) <-> P(z)). Therefore, the answer is no.