Consider the following premises. For all x_17, if x_17 makes tea, then x_17 is not excited. For all x_17, x_17 is a janitor or x_17 is excited. For all x_17, if x_17 cooks, then x_17 is a writer. For all x_17, if x_17 were a writer, then x_17 is making tea. There is at least one x_17 for which x_17 is not a janitor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_17 for which x_17 will not cook.
 From the fact that for all x_17, x_17 is a janitor or x_17 is excited, and that there is at least one x_17 for which x_17 is not a janitor we can infer that there is at least one x_17 for which x_17 is excited via existential disjunctive syllogism. From the fact that for all x_17, if x_17 cooks, then x_17 is a writer, and that for all x_17, if x_17 were a writer, then x_17 is making tea we can infer that for all x_17, if x_17 cooks, then x_17 is making tea via universal transitivity. From the fact that for all x_17, if x_17 makes tea, then x_17 is not excited, and that there is at least one x_17 for which x_17 is excited we can infer that there is at least one x_17 for which x_17 will not make tea via existential modus tollens. Finally, from the fact that for all x_17, if x_17 cooks, then x_17 is making tea, and that there is at least one x_17 for which x_17 will not make tea we can infer that there is at least one x_17 for which x_17 will not cook via existential modus tollens. Therefore, the answer is yes.