Consider the following premises. For all x_8, x_8 will write a letter or x_8 will listen to a song. For all x_8, x_8 will make tea. There is at least one x_8 for which if x_8 sleeps, then x_8 is not playing squash. For all x_8, if x_8 listens to a song, then x_8 is not making tea. For all x_8, x_8 will not write a letter or x_8 will play squash. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_8 for which x_8 will not sleep.
 From the fact that for all x_8, if x_8 listens to a song, then x_8 is not making tea, and that for all x_8, x_8 will make tea we can infer that for all x_8, x_8 will not listen to a song via universal modus tollens. From the fact that for all x_8, x_8 will write a letter or x_8 will listen to a song, and that for all x_8, x_8 will not write a letter or x_8 will play squash we can infer that for all x_8, x_8 will listen to a song or x_8 will play squash via universal resolution. From the fact that for all x_8, x_8 will listen to a song or x_8 will play squash, and that for all x_8, x_8 will not listen to a song we can infer that for all x_8, x_8 will play squash via universal disjunctive syllogism. Finally, from the fact that there is at least one x_8 for which if x_8 sleeps, then x_8 is not playing squash, and that for all x_8, x_8 will play squash we can infer that there is at least one x_8 for which x_8 will not sleep via existential modus tollens. Therefore, the answer is yes.