Consider the following premises. forall x_8: P_2(x_8) or P(x_8). forall x_8: ~R_8(x_8). forall x_8: R_8(x_8) -> P(x_8). forall x_8: Q_8(x_8) -> R_8(x_8). forall x_8: P(x_8) -> Q_8(x_8). forall x_8: ~P_2(x_8) or R_2(x_8). Can we infer forall x_8: R_2(x_8) from them? If possible, name the inference rules being used at each step.
 From forall x_8: P(x_8) -> Q_8(x_8), forall x_8: Q_8(x_8) -> R_8(x_8) we can infer forall x_8: P(x_8) -> R_8(x_8) via universal transitivity. From forall x_8: P(x_8) -> R_8(x_8), forall x_8: R_8(x_8) -> P(x_8) we can infer forall x_8: P(x_8) <-> R_8(x_8) via universal biconditional introduction. From forall x_8: P(x_8) <-> R_8(x_8), forall x_8: ~R_8(x_8) we can infer forall x_8: ~P(x_8) via universal biconditional elimination. From forall x_8: P_2(x_8) or P(x_8), forall x_8: ~P_2(x_8) or R_2(x_8) we can infer forall x_8: P(x_8) or R_2(x_8) via universal resolution. Finally, from forall x_8: P(x_8) or R_2(x_8), forall x_8: ~P(x_8) we can infer forall x_8: R_2(x_8) via universal disjunctive syllogism. Therefore, the answer is yes.