Consider the following premises. forall x_5: P_5(x_5) or ~T_4(x_5). forall x_5: ~P_2(x_5) or R_2(x_5). forall x_5: R_2(x_5) -> P_5(x_5). exists x_5: P_2(x_5) or T_4(x_5). Can we infer exists x_5: P_5(x_5) from them? If possible, name the inference rules being used at each step.
 From forall x_5: P_5(x_5) or ~T_4(x_5) we can infer forall x_5: T_4(x_5) -> P_5(x_5) via universal rewriting. From exists x_5: P_2(x_5) or T_4(x_5), forall x_5: ~P_2(x_5) or R_2(x_5) we can infer exists x_5: T_4(x_5) or R_2(x_5) via existential resolution. Finally, from forall x_5: T_4(x_5) -> P_5(x_5), forall x_5: R_2(x_5) -> P_5(x_5), exists x_5: T_4(x_5) or R_2(x_5) we can infer exists x_5: P_5(x_5) via existential disjunction elimination. Therefore, the answer is yes.