Consider the following premises. forall x_13: P_13(x_13) or Q_13(x_13). forall x_13: P_13(x_13) -> R_13(x_13). r_1 -> ~R_13(j). forall x_13: Q_13(x_13) -> R_13(x_13). Can we infer r_1 from them? If possible, name the inference rules being used at each step.
 From forall x_13: P_13(x_13) -> R_13(x_13), forall x_13: Q_13(x_13) -> R_13(x_13), forall x_13: P_13(x_13) or Q_13(x_13) we can infer forall x_13: R_13(x_13) via universal disjunction elimination. From forall x_13: R_13(x_13) we can infer R_13(j) via universal instantiation. Finally, from r_1 -> ~R_13(j), R_13(j) we can infer ~r_1 via modus tollens, which contradicts r_1. Therefore, the answer is no.