Consider the following premises. For all x_7, x_7 will play a game. There is at least one x_7 for which x_7 will cook or x_7 is not a janitor. For all x_7, if x_7 cooks, then x_7 is not playing a game. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_7 for which x_7 is not a janitor.
 From the fact that for all x_7, if x_7 cooks, then x_7 is not playing a game, and that for all x_7, x_7 will play a game we can infer that for all x_7, x_7 will not cook via universal modus tollens. From the fact that there is at least one x_7 for which x_7 will cook or x_7 is not a janitor we can infer that there is at least one x_7 for which if x_7 were a janitor, then x_7 is cooking via existential rewriting. Finally, from the fact that there is at least one x_7 for which if x_7 were a janitor, then x_7 is cooking, and that for all x_7, x_7 will not cook we can infer that there is at least one x_7 for which x_7 is not a janitor via existential modus tollens. Therefore, the answer is yes.