Consider the following premises. forall z: R_2(z) <-> R_1(z). forall z: R_1(z) <-> U(z). Can we infer forall z: U(z) -> R_2(z) from them? If possible, name the inference rules being used at each step.
 From forall z: R_2(z) <-> R_1(z) we can infer forall z: R_1(z) -> R_2(z) via universal biconditional elimination. From forall z: R_1(z) <-> U(z) we can infer forall z: U(z) -> R_1(z) via universal biconditional elimination. Finally, from forall z: U(z) -> R_1(z), forall z: R_1(z) -> R_2(z) we can infer forall z: U(z) -> R_2(z) via universal transitivity. Therefore, the answer is yes.