Consider the following premises. exists x_5: R_2(x_5) -> U(x_5). forall x_5: U(x_5) -> P_2(x_5). forall x_5: ~P_2(x_5). forall x_5: U(x_5) -> R_2(x_5). Can we infer exists x_5: ~R_2(x_5) from them? If possible, name the inference rules being used at each step.
 From exists x_5: R_2(x_5) -> U(x_5), forall x_5: U(x_5) -> R_2(x_5) we can infer exists x_5: R_2(x_5) <-> U(x_5) via existential biconditional introduction. From forall x_5: U(x_5) -> P_2(x_5), forall x_5: ~P_2(x_5) we can infer forall x_5: ~U(x_5) via universal modus tollens. Finally, from exists x_5: R_2(x_5) <-> U(x_5), forall x_5: ~U(x_5) we can infer exists x_5: ~R_2(x_5) via existential biconditional elimination. Therefore, the answer is yes.