Consider the following premises. forall z: W_3(z) <-> R_4(z). forall z: W_3(z) -> R_4(z). Can we infer exists z: ~(W_3(z) -> R_4(z)) from them? If possible, name the inference rules being used at each step.
 From forall z: W_3(z) <-> R_4(z) we can infer forall z: R_4(z) -> W_3(z) via universal biconditional elimination. From forall z: R_4(z) -> W_3(z), forall z: W_3(z) -> R_4(z) we can infer forall z: R_4(z) <-> W_3(z) via universal biconditional introduction. Finally, from forall z: R_4(z) <-> W_3(z) we can infer forall z: W_3(z) -> R_4(z) via universal biconditional elimination, which contradicts exists z: ~(W_3(z) -> R_4(z)). Therefore, the answer is no.