Consider the following premises. For all x_9, x_9 is a cashier. There is at least one x_9 for which if x_9 were excited, then x_9 is not a soldier. For all x_9, if x_9 were a cashier, then x_9 is a soldier. For all x_9, x_9 is excited or x_9 will cook. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_9 for which x_9 will cook.
 From the fact that for all x_9, if x_9 were a cashier, then x_9 is a soldier, and that for all x_9, x_9 is a cashier we can infer that for all x_9, x_9 is a soldier via universal modus ponens. From the fact that there is at least one x_9 for which if x_9 were excited, then x_9 is not a soldier, and that for all x_9, x_9 is a soldier we can infer that there is at least one x_9 for which x_9 is not excited via existential modus tollens. Finally, from the fact that for all x_9, x_9 is excited or x_9 will cook, and that there is at least one x_9 for which x_9 is not excited we can infer that there is at least one x_9 for which x_9 will cook via existential disjunctive syllogism. Therefore, the answer is yes.