Consider the following premises. forall z_2: R(z_2) -> T(z_2). forall z_2: T(z_2) -> U(z_2). forall z_2: R_2(z_2) -> W(z_2). forall z_2: R_2(z_2) -> R(z_2). forall z_2: W(z_2) -> R_2(z_2). Can we infer forall z_2: W(z_2) -> U(z_2) from them? If possible, name the inference rules being used at each step.
 From forall z_2: R(z_2) -> T(z_2), forall z_2: T(z_2) -> U(z_2) we can infer forall z_2: R(z_2) -> U(z_2) via universal transitivity. From forall z_2: R_2(z_2) -> W(z_2), forall z_2: W(z_2) -> R_2(z_2) we can infer forall z_2: R_2(z_2) <-> W(z_2) via universal biconditional introduction. From forall z_2: R_2(z_2) <-> W(z_2) we can infer forall z_2: W(z_2) -> R_2(z_2) via universal biconditional elimination. From forall z_2: W(z_2) -> R_2(z_2), forall z_2: R_2(z_2) -> R(z_2) we can infer forall z_2: W(z_2) -> R(z_2) via universal transitivity. Finally, from forall z_2: W(z_2) -> R(z_2), forall z_2: R(z_2) -> U(z_2) we can infer forall z_2: W(z_2) -> U(z_2) via universal transitivity. Therefore, the answer is yes.