Consider the following premises. There is at least one x_2 for which if x_2 plays a game, then x_2 is writing a letter. For all x_2, if x_2 reads a book, then x_2 is writing a letter. For all x_2, x_2 will read a book or x_2 will make tea. For all x_2, if x_2 makes tea, then x_2 is playing a game. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_2 for which x_2 will write a letter.
 From the fact that for all x_2, if x_2 makes tea, then x_2 is playing a game, and that there is at least one x_2 for which if x_2 plays a game, then x_2 is writing a letter we can infer that there is at least one x_2 for which if x_2 makes tea, then x_2 is writing a letter via existential transitivity. Finally, from the fact that for all x_2, if x_2 reads a book, then x_2 is writing a letter, that there is at least one x_2 for which if x_2 makes tea, then x_2 is writing a letter, and that for all x_2, x_2 will read a book or x_2 will make tea we can infer that there is at least one x_2 for which x_2 will write a letter via existential disjunction elimination. Therefore, the answer is yes.