Consider the following premises. For all x_6, x_6 will listen to a song or x_6 will not sleep. For all x_6, x_6 will play a game. For all x_6, if x_6 listens to a song, then x_6 is not playing a game. Can we infer the following from them? If we can, name the inference rule being used: For all x_6, x_6 will not sleep.
 From the fact that for all x_6, x_6 will listen to a song or x_6 will not sleep we can infer that for all x_6, if x_6 sleeps, then x_6 is listening to a song via universal rewriting. From the fact that for all x_6, if x_6 listens to a song, then x_6 is not playing a game, and that for all x_6, x_6 will play a game we can infer that for all x_6, x_6 will not listen to a song via universal modus tollens. Finally, from the fact that for all x_6, if x_6 sleeps, 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 sleep via universal modus tollens. Therefore, the answer is yes.