Consider the following premises. exists z: W(z) or R(z). forall z: U_1(z) -> U(z). forall z: R(z) -> U_1(z). forall z: U(z) <-> W(z). Can we infer exists z: U(z) from them? If possible, name the inference rules being used at each step.
 From forall z: U(z) <-> W(z) we can infer forall z: W(z) -> U(z) via universal biconditional elimination. From forall z: R(z) -> U_1(z), forall z: U_1(z) -> U(z) we can infer forall z: R(z) -> U(z) via universal transitivity. Finally, from forall z: W(z) -> U(z), forall z: R(z) -> U(z), exists z: W(z) or R(z) we can infer exists z: U(z) via existential disjunction elimination. Therefore, the answer is yes.