Consider the following premises. forall z: T(z) <-> W_2(z). forall z: R(z) <-> W_2(z). forall z: T(z) -> R(z). Can we infer exists z: ~(W_2(z) -> R(z)) from them? If possible, name the inference rules being used at each step.
 From forall z: T(z) <-> W_2(z) we can infer forall z: W_2(z) -> T(z) via universal biconditional elimination. From forall z: W_2(z) -> T(z), forall z: T(z) -> R(z) we can infer forall z: W_2(z) -> R(z) via universal transitivity. From forall z: R(z) <-> W_2(z) we can infer forall z: R(z) -> W_2(z) via universal biconditional elimination. From forall z: R(z) -> W_2(z), forall z: W_2(z) -> R(z) we can infer forall z: R(z) <-> W_2(z) via universal biconditional introduction. Finally, from forall z: R(z) <-> W_2(z) we can infer forall z: W_2(z) -> R(z) via universal biconditional elimination, which contradicts exists z: ~(W_2(z) -> R(z)). Therefore, the answer is no.