Consider the following premises. For all x_6, x_6 will not listen to a song. For all x_6, x_6 will listen to a song or x_6 is a doctor. For all x_6, x_6 will sleep or x_6 is not a doctor. Can we infer the following from them? If we can, name the inference rule being used: For all x_6, x_6 will sleep.
 From the fact that for all x_6, x_6 will listen to a song or x_6 is a doctor, and that for all x_6, x_6 will not listen to a song we can infer that for all x_6, x_6 is a doctor via universal disjunctive syllogism. From the fact that for all x_6, x_6 will sleep or x_6 is not a doctor we can infer that for all x_6, if x_6 were a doctor, then x_6 is sleeping via universal rewriting. Finally, from the fact that for all x_6, if x_6 were a doctor, then x_6 is sleeping, and that for all x_6, x_6 is a doctor we can infer that for all x_6, x_6 will sleep via universal modus ponens. Therefore, the answer is yes.