Consider the following premises. For all x_13, if x_13 were excited, then x_13 is sleeping. There is at least one x_13 for which x_13 will listen to a song if and only if x_13 will sleep. For all x_13, x_13 will listen to a song or x_13 is excited. For all x_13, if x_13 were a cashier, then x_13 is not sleeping. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_13 for which x_13 is not a cashier.
 From the fact that there is at least one x_13 for which x_13 will listen to a song if and only if x_13 will sleep we can infer that there is at least one x_13 for which if x_13 listens to a song, then x_13 is sleeping via existential biconditional elimination. From the fact that there is at least one x_13 for which if x_13 listens to a song, then x_13 is sleeping, that for all x_13, if x_13 were excited, then x_13 is sleeping, and that for all x_13, x_13 will listen to a song or x_13 is excited we can infer that there is at least one x_13 for which x_13 will sleep via existential disjunction elimination. Finally, from the fact that for all x_13, if x_13 were a cashier, then x_13 is not sleeping, and that there is at least one x_13 for which x_13 will sleep we can infer that there is at least one x_13 for which x_13 is not a cashier via existential modus tollens. Therefore, the answer is yes.