Consider the following premises. forall z: P(z) -> U_3(z). forall z: U_3(z) -> W_3(z). forall z: W_3(z) -> P(z). Can we infer exists z: ~(U_3(z) -> W_3(z)) from them? If possible, name the inference rules being used at each step.
 From forall z: W_3(z) -> P(z), forall z: P(z) -> U_3(z) we can infer forall z: W_3(z) -> U_3(z) via universal transitivity. From forall z: U_3(z) -> W_3(z), forall z: W_3(z) -> U_3(z) we can infer forall z: U_3(z) <-> W_3(z) via universal biconditional introduction. Finally, from forall z: U_3(z) <-> W_3(z) we can infer forall z: U_3(z) -> W_3(z) via universal biconditional elimination, which contradicts exists z: ~(U_3(z) -> W_3(z)). Therefore, the answer is no.