Consider the following premises. forall z_2: ~W_4(z_2). exists z_2: R(z_2) <-> S_4(z_2). forall z_2: S_4(z_2) <-> T(z_2). forall z_2: T(z_2) <-> W_4(z_2). Can we infer exists z_2: ~R(z_2) from them? If possible, name the inference rules being used at each step.
 From forall z_2: T(z_2) <-> W_4(z_2), forall z_2: ~W_4(z_2) we can infer forall z_2: ~T(z_2) via universal biconditional elimination. From forall z_2: S_4(z_2) <-> T(z_2), forall z_2: ~T(z_2) we can infer forall z_2: ~S_4(z_2) via universal biconditional elimination. Finally, from exists z_2: R(z_2) <-> S_4(z_2), forall z_2: ~S_4(z_2) we can infer exists z_2: ~R(z_2) via existential biconditional elimination. Therefore, the answer is yes.