Consider the following premises. For all x_10, x_10 is a cashier or x_10 will listen to a song. For all x_10, x_10 will not sleep. For all x_10, if x_10 were a cashier, then x_10 is sleeping. There is at least one x_10 for which x_10 will listen to a song if and only if x_10 is an artist. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_10 for which x_10 is an artist.
 From the fact that for all x_10, if x_10 were a cashier, then x_10 is sleeping, and that for all x_10, x_10 will not sleep we can infer that for all x_10, x_10 is not a cashier via universal modus tollens. From the fact that there is at least one x_10 for which x_10 will listen to a song if and only if x_10 is an artist we can infer that there is at least one x_10 for which if x_10 listens to a song, then x_10 is an artist via existential biconditional elimination. From the fact that for all x_10, x_10 is a cashier or x_10 will listen to a song, and that for all x_10, x_10 is not a cashier we can infer that for all x_10, x_10 will listen to a song via universal disjunctive syllogism. Finally, from the fact that there is at least one x_10 for which if x_10 listens to a song, then x_10 is an artist, and that for all x_10, x_10 will listen to a song we can infer that there is at least one x_10 for which x_10 is an artist via existential modus ponens. Therefore, the answer is yes.