Consider the following premises. forall z: R(z) or ~S_2(z). exists z: R_3(z) <-> R(z). forall z: R_3(z) -> S_2(z). Can we infer exists z: S_2(z) <-> R(z) from them? If possible, name the inference rules being used at each step.
 From exists z: R_3(z) <-> R(z) we can infer exists z: R(z) -> R_3(z) via existential biconditional elimination. From forall z: R(z) or ~S_2(z) we can infer forall z: S_2(z) -> R(z) via universal rewriting. From exists z: R(z) -> R_3(z), forall z: R_3(z) -> S_2(z) we can infer exists z: R(z) -> S_2(z) via existential transitivity. Finally, from forall z: S_2(z) -> R(z), exists z: R(z) -> S_2(z) we can infer exists z: S_2(z) <-> R(z) via existential biconditional introduction. Therefore, the answer is yes.