Consider the following premises. For all x_6, if x_6 were a writer, then x_6 is drinking water. For all x_6, if x_6 sleeps, then x_6 is drinking water. There is at least one x_6 for which x_6 will listen to a song. For all x_6, if x_6 listens to a song, then x_6 is sleeping. 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 drink water.
 From the fact that for all x_6, if x_6 listens to a song, then x_6 is sleeping, and that there is at least one x_6 for which x_6 will listen to a song we can infer that there is at least one x_6 for which x_6 will sleep via existential modus ponens. From the fact that there is at least one x_6 for which x_6 will sleep we can infer that there is at least one x_6 for which x_6 will sleep or x_6 is a writer via existential addition. Finally, from the fact that for all x_6, if x_6 sleeps, then x_6 is drinking water, that for all x_6, if x_6 were a writer, then x_6 is drinking water, and that there is at least one x_6 for which x_6 will sleep or x_6 is a writer we can infer that there is at least one x_6 for which x_6 will drink water via existential disjunction elimination. Therefore, the answer is yes.