Consider the following premises. forall z: ~R_3(z) or W(z). forall z: R(z) -> R_2(z). exists z: R_2(z) or ~W(z). forall z: R_3(z) or R(z). Can we infer exists z: R_2(z) from them? If possible, name the inference rules being used at each step.
 From forall z: R_3(z) or R(z), forall z: ~R_3(z) or W(z) we can infer forall z: R(z) or W(z) via universal resolution. From exists z: R_2(z) or ~W(z) we can infer exists z: W(z) -> R_2(z) via existential rewriting. Finally, from forall z: R(z) -> R_2(z), exists z: W(z) -> R_2(z), forall z: R(z) or W(z) we can infer exists z: R_2(z) via existential disjunction elimination. Therefore, the answer is yes.