Consider the following premises. For all x_6, if x_6 plays tennis, then x_6 is a janitor. There is at least one x_6 for which if x_6 were a janitor, then x_6 is playing tennis. For all x_6, if x_6 cooks, then x_6 is a janitor. For all x_6, x_6 will play tennis or x_6 will cook. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_6 for which x_6 is a janitor.
 From the fact that there is at least one x_6 for which if x_6 were a janitor, then x_6 is playing tennis, and that for all x_6, if x_6 plays tennis, then x_6 is a janitor we can infer that there is at least one x_6 for which x_6 is a janitor if and only if x_6 will play tennis via existential biconditional introduction. From the fact that there is at least one x_6 for which x_6 is a janitor if and only if x_6 will play tennis we can infer that there is at least one x_6 for which if x_6 plays tennis, then x_6 is a janitor via existential biconditional elimination. Finally, from the fact that there is at least one x_6 for which if x_6 plays tennis, then x_6 is a janitor, that for all x_6, if x_6 cooks, then x_6 is a janitor, and that for all x_6, x_6 will play tennis or x_6 will cook we can infer that there is at least one x_6 for which x_6 is a janitor via existential disjunction elimination. Therefore, the answer is yes.