Consider the following premises. For all x_2, if x_2 cooks, then x_2 is running. There is at least one x_2 for which x_2 will not go running. For all x_2, if x_2 were a bartender, then x_2 is cooking. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_2 for which x_2 is not a bartender.
 From the fact that for all x_2, if x_2 cooks, then x_2 is running, and that there is at least one x_2 for which x_2 will not go running we can infer that there is at least one x_2 for which x_2 will not cook via existential modus tollens. Finally, from the fact that for all x_2, if x_2 were a bartender, then x_2 is cooking, and that there is at least one x_2 for which x_2 will not cook we can infer that there is at least one x_2 for which x_2 is not a bartender via existential modus tollens. Therefore, the answer is yes.