Consider the following premises. For all y_1, if y_1 were a janitor, then y_1 is reading a book. For all y_1, if y_1 cooks, then y_1 is playing squash. There is at least one y_1 for which if y_1 reads a book, then y_1 is playing squash. For all y_1, y_1 is a janitor or y_1 will cook. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y_1 for which y_1 will play squash.
 From the fact that for all y_1, if y_1 were a janitor, then y_1 is reading a book, and that there is at least one y_1 for which if y_1 reads a book, then y_1 is playing squash we can infer that there is at least one y_1 for which if y_1 were a janitor, then y_1 is playing squash via existential transitivity. Finally, from the fact that there is at least one y_1 for which if y_1 were a janitor, then y_1 is playing squash, that for all y_1, if y_1 cooks, then y_1 is playing squash, and that for all y_1, y_1 is a janitor or y_1 will cook we can infer that there is at least one y_1 for which y_1 will play squash via existential disjunction elimination. Therefore, the answer is yes.