Consider the following premises. forall x_11: P_5(x_11) or Q_5(x_11). forall x_11: Q_11(x_11) -> R_11(x_11). forall x_11: ~P_5(x_11). forall x_11: P_11(x_11) or Q_11(x_11). forall x_11: P_11(x_11) -> R_11(x_11). Can we infer R_11(j_1) from them? If possible, name the inference rules being used at each step.
 From forall x_11: P_11(x_11) -> R_11(x_11), forall x_11: Q_11(x_11) -> R_11(x_11), forall x_11: P_11(x_11) or Q_11(x_11) we can infer forall x_11: R_11(x_11) via universal disjunction elimination. From forall x_11: P_5(x_11) or Q_5(x_11), forall x_11: ~P_5(x_11) we can infer forall x_11: Q_5(x_11) via universal disjunctive syllogism. From forall x_11: R_11(x_11), forall x_11: Q_5(x_11) we can infer forall x_11: R_11(x_11) and Q_5(x_11) via universal conjunction. From forall x_11: R_11(x_11) and Q_5(x_11) we can infer R_11(j_1) and Q_5(j_1) via universal instantiation. Finally, from R_11(j_1) and Q_5(j_1) we can infer R_11(j_1) via simplification. Therefore, the answer is yes.