Consider the following premises. For all x_4, x_4 will cook or x_4 is a soldier. For all x_4, if x_4 cooks, then x_4 is not making tea. There is at least one x_4 for which x_4 will make tea. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_4 for which x_4 is a soldier.
 From the fact that for all x_4, if x_4 cooks, then x_4 is not making tea, and that there is at least one x_4 for which x_4 will make tea we can infer that there is at least one x_4 for which x_4 will not cook via existential modus tollens. Finally, from the fact that for all x_4, x_4 will cook or x_4 is a soldier, and that there is at least one x_4 for which x_4 will not cook we can infer that there is at least one x_4 for which x_4 is a soldier via existential disjunctive syllogism. Therefore, the answer is yes.