Consider the following premises. forall z: R_2(z) -> U_2(z). forall z: P(z) or R_2(z). forall z: U_2(z) -> Q(z). exists z: Q(z) <-> P(z). Can we infer exists z: Q(z) from them? If possible, name the inference rules being used at each step.
 From forall z: R_2(z) -> U_2(z), forall z: U_2(z) -> Q(z) we can infer forall z: R_2(z) -> Q(z) via universal transitivity. From exists z: Q(z) <-> P(z) we can infer exists z: P(z) -> Q(z) via existential biconditional elimination. Finally, from exists z: P(z) -> Q(z), forall z: R_2(z) -> Q(z), forall z: P(z) or R_2(z) we can infer exists z: Q(z) via existential disjunction elimination. Therefore, the answer is yes.