Consider the following premises. For all x_2, if x_2 were a soldier, then x_2 is reading a book. There is at least one x_2 for which if x_2 reads a book, then x_2 is a bartender. For all x_2, if x_2 writes a letter, then x_2 is a bartender. For all x_2, x_2 will write a letter or x_2 is a soldier. 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 is a bartender.
 From the fact that for all x_2, if x_2 were a soldier, then x_2 is reading a book, and that there is at least one x_2 for which if x_2 reads a book, then x_2 is a bartender we can infer that there is at least one x_2 for which if x_2 were a soldier, then x_2 is a bartender via existential transitivity. Finally, from the fact that for all x_2, if x_2 writes a letter, then x_2 is a bartender, that there is at least one x_2 for which if x_2 were a soldier, then x_2 is a bartender, and that for all x_2, x_2 will write a letter or x_2 is a soldier we can infer that there is at least one x_2 for which x_2 is a bartender via existential disjunction elimination. Therefore, the answer is yes.