Consider the following premises. For all x_6, if x_6 writes a letter, then x_6 is sleeping. For all x_6, x_6 will not listen to a song. For all x_6, if x_6 writes a letter, then x_6 is listening to a song. There is at least one x_6 for which if x_6 sleeps, then x_6 is writing a letter. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_6 for which x_6 will not sleep.
 From the fact that for all x_6, if x_6 writes a letter, then x_6 is sleeping, and that there is at least one x_6 for which if x_6 sleeps, then x_6 is writing a letter we can infer that there is at least one x_6 for which x_6 will write a letter if and only if x_6 will sleep via existential biconditional introduction. From the fact that for all x_6, if x_6 writes a letter, then x_6 is listening to a song, and that for all x_6, x_6 will not listen to a song we can infer that for all x_6, x_6 will not write a letter via universal modus tollens. Finally, from the fact that there is at least one x_6 for which x_6 will write a letter if and only if x_6 will sleep, and that for all x_6, x_6 will not write a letter we can infer that there is at least one x_6 for which x_6 will not sleep via existential biconditional elimination. Therefore, the answer is yes.