Consider the following premises. exists x_5: P_5(x_5) or ~R_3(x_5). forall x_5: ~P_5(x_5). forall x_5: P_5(x_5) -> R_3(x_5). Can we infer exists x_5: ~R_3(x_5) from them? If possible, name the inference rules being used at each step.
 From exists x_5: P_5(x_5) or ~R_3(x_5) we can infer exists x_5: R_3(x_5) -> P_5(x_5) via existential rewriting. From exists x_5: R_3(x_5) -> P_5(x_5), forall x_5: P_5(x_5) -> R_3(x_5) we can infer exists x_5: R_3(x_5) <-> P_5(x_5) via existential biconditional introduction. Finally, from exists x_5: R_3(x_5) <-> P_5(x_5), forall x_5: ~P_5(x_5) we can infer exists x_5: ~R_3(x_5) via existential biconditional elimination. Therefore, the answer is yes.