Consider the following premises. For all x_6, x_6 will read a book or x_6 is excited. For all x_6, if x_6 were excited, then x_6 is playing a game. There is at least one x_6 for which x_6 is a janitor or x_6 will not read a book. For all x_6, if x_6 plays a game, then x_6 is a janitor. 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 for all x_6, if x_6 were excited, then x_6 is playing a game, and that for all x_6, if x_6 plays a game, then x_6 is a janitor we can infer that for all x_6, if x_6 were excited, then x_6 is a janitor via universal transitivity. From the fact that there is at least one x_6 for which x_6 is a janitor or x_6 will not read a book we can infer that there is at least one x_6 for which if x_6 reads a book, then x_6 is a janitor via existential rewriting. Finally, from the fact that there is at least one x_6 for which if x_6 reads a book, then x_6 is a janitor, that for all x_6, if x_6 were excited, then x_6 is a janitor, and that for all x_6, x_6 will read a book or x_6 is excited 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.