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