Consider the following premises. forall z: R_4(z) <-> W_2(z). forall z: W(z) <-> W_2(z). forall z: ~W(z). Can we infer exists z: R_4(z) from them? If possible, name the inference rules being used at each step.
 From forall z: W(z) <-> W_2(z), forall z: ~W(z) we can infer forall z: ~W_2(z) via universal biconditional elimination. From forall z: R_4(z) <-> W_2(z) we can infer forall z: R_4(z) -> W_2(z) via universal biconditional elimination. Finally, from forall z: R_4(z) -> W_2(z), forall z: ~W_2(z) we can infer forall z: ~R_4(z) via universal modus tollens, which contradicts exists z: R_4(z). Therefore, the answer is no.