Consider the following premises. forall z_3: R(z_3) <-> T(z_3). forall z_3: R(z_3) -> W_4(z_3). forall z_3: W_2(z_3) -> T(z_3). forall z_3: W_4(z_3) -> W_2(z_3). Can we infer exists z_3: ~(R(z_3) <-> W_4(z_3)) from them? If possible, name the inference rules being used at each step.
 From forall z_3: W_4(z_3) -> W_2(z_3), forall z_3: W_2(z_3) -> T(z_3) we can infer forall z_3: W_4(z_3) -> T(z_3) via universal transitivity. From forall z_3: R(z_3) <-> T(z_3) we can infer forall z_3: T(z_3) -> R(z_3) via universal biconditional elimination. From forall z_3: W_4(z_3) -> T(z_3), forall z_3: T(z_3) -> R(z_3) we can infer forall z_3: W_4(z_3) -> R(z_3) via universal transitivity. Finally, from forall z_3: R(z_3) -> W_4(z_3), forall z_3: W_4(z_3) -> R(z_3) we can infer forall z_3: R(z_3) <-> W_4(z_3) via universal biconditional introduction, which contradicts exists z_3: ~(R(z_3) <-> W_4(z_3)). Therefore, the answer is no.