Consider the following premises. forall x_13: P_13(x_13) -> U(x_13). forall x_13: R(x_13) -> ~U(x_13). R(a_5) or w. forall x_13: Q_13(x_13) -> U(x_13). forall x_13: P_13(x_13) or Q_13(x_13). Can we infer w from them? If possible, name the inference rules being used at each step.
 From forall x_13: P_13(x_13) -> U(x_13), forall x_13: Q_13(x_13) -> U(x_13), forall x_13: P_13(x_13) or Q_13(x_13) we can infer forall x_13: U(x_13) via universal disjunction elimination. From forall x_13: R(x_13) -> ~U(x_13), forall x_13: U(x_13) we can infer forall x_13: ~R(x_13) via universal modus tollens. From forall x_13: ~R(x_13) we can infer ~R(a_5) via universal instantiation. Finally, from R(a_5) or w, ~R(a_5) we can infer w via disjunctive syllogism. Therefore, the answer is yes.