Consider the following premises. forall x_12: P_12(x_12) or ~Q_12(x_12). forall x_12: P_9(x_12) -> R_5(x_12). forall x_12: U(x_12) -> ~R_5(x_12). forall x_12: P_12(x_12) -> R_5(x_12). forall x_12: P_9(x_12). Can we infer forall x_12: ~U(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) or ~Q_12(x_12) we can infer forall x_12: Q_12(x_12) -> P_12(x_12) via universal rewriting. From forall x_12: P_9(x_12) we can infer forall x_12: P_9(x_12) or Q_12(x_12) via universal addition. From forall x_12: Q_12(x_12) -> P_12(x_12), forall x_12: P_12(x_12) -> R_5(x_12) we can infer forall x_12: Q_12(x_12) -> R_5(x_12) via universal transitivity. From forall x_12: P_9(x_12) -> R_5(x_12), forall x_12: Q_12(x_12) -> R_5(x_12), forall x_12: P_9(x_12) or Q_12(x_12) we can infer forall x_12: R_5(x_12) via universal disjunction elimination. Finally, from forall x_12: U(x_12) -> ~R_5(x_12), forall x_12: R_5(x_12) we can infer forall x_12: ~U(x_12) via universal modus tollens. Therefore, the answer is yes.