Consider the following premises. For all x_1, if x_1 were a soldier, then x_1 is sleeping. For all x_1, x_1 is a nurse or x_1 is a soldier. There is at least one x_1 for which if x_1 were a nurse, then x_1 is running. For all x_1, if x_1 goes running, then x_1 is sleeping. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_1 for which x_1 will sleep.
 From the fact that there is at least one x_1 for which if x_1 were a nurse, then x_1 is running, and that for all x_1, if x_1 goes running, then x_1 is sleeping we can infer that there is at least one x_1 for which if x_1 were a nurse, then x_1 is sleeping via existential transitivity. Finally, from the fact that there is at least one x_1 for which if x_1 were a nurse, then x_1 is sleeping, that for all x_1, if x_1 were a soldier, then x_1 is sleeping, and that for all x_1, x_1 is a nurse or x_1 is a soldier we can infer that there is at least one x_1 for which x_1 will sleep via existential disjunction elimination. Therefore, the answer is yes.