Consider the following premises. For all x_1, if x_1 listens to a song, then x_1 is a cashier. For all x_1, x_1 is a writer. There is at least one x_1 for which if x_1 were a writer, then x_1 is a cashier. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_1 for which x_1 is a cashier.
 From the fact that for all x_1, x_1 is a writer we can infer that for all x_1, x_1 is a writer or x_1 will listen to a song via universal addition. Finally, from the fact that there is at least one x_1 for which if x_1 were a writer, then x_1 is a cashier, that for all x_1, if x_1 listens to a song, then x_1 is a cashier, and that for all x_1, x_1 is a writer or x_1 will listen to a song we can infer that there is at least one x_1 for which x_1 is a cashier via existential disjunction elimination. Therefore, the answer is yes.