Consider the following premises. forall z: R(z) -> U(z). exists z: ~U(z). forall z: U_1(z) -> R(z). forall z: U(z) -> U_1(z). Can we infer forall z: R(z) from them? If possible, name the inference rules being used at each step.
 From forall z: U(z) -> U_1(z), forall z: U_1(z) -> R(z) we can infer forall z: U(z) -> R(z) via universal transitivity. From forall z: R(z) -> U(z), forall z: U(z) -> R(z) we can infer forall z: R(z) <-> U(z) via universal biconditional introduction. Finally, from forall z: R(z) <-> U(z), exists z: ~U(z) we can infer exists z: ~R(z) via existential biconditional elimination, which contradicts forall z: R(z). Therefore, the answer is no.