Consider the following premises. For all x_7, if x_7 cooks, then x_7 is driving a car. There is at least one x_7 for which x_7 will cook or x_7 is fast. For all x_7, x_7 is a janitor if and only if x_7 will drive a car. For all x_7, x_7 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_7 for which x_7 is fast.
 From the fact that for all x_7, x_7 is a janitor if and only if x_7 will drive a car, and that for all x_7, x_7 is not a janitor we can infer that for all x_7, x_7 will not drive a car via universal biconditional elimination. From the fact that for all x_7, if x_7 cooks, then x_7 is driving a car, and that for all x_7, x_7 will not drive a car we can infer that for all x_7, x_7 will not cook via universal modus tollens. Finally, from the fact that there is at least one x_7 for which x_7 will cook or x_7 is fast, 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 fast via existential disjunctive syllogism. Therefore, the answer is yes.