Consider the following premises. forall z: U(z) -> S_1(z). forall z: R_1(z) <-> S_1(z). forall z: P(z) -> U(z). forall z: ~R_1(z). forall z: S_1(z) -> P(z). Can we infer forall z: ~U(z) from them? If possible, name the inference rules being used at each step.
 From forall z: S_1(z) -> P(z), forall z: P(z) -> U(z) we can infer forall z: S_1(z) -> U(z) via universal transitivity. From forall z: S_1(z) -> U(z), forall z: U(z) -> S_1(z) we can infer forall z: S_1(z) <-> U(z) via universal biconditional introduction. From forall z: R_1(z) <-> S_1(z), forall z: ~R_1(z) we can infer forall z: ~S_1(z) via universal biconditional elimination. Finally, from forall z: S_1(z) <-> U(z), forall z: ~S_1(z) we can infer forall z: ~U(z) via universal biconditional elimination. Therefore, the answer is yes.