Consider the following premises. For all x_5, if x_5 climbs a mountain, then x_5 is sleeping. For all x_5, x_5 is an astronaut. For all x_5, x_5 will sleep if and only if x_5 is an astronaut. Can we infer the following from them? If we can, name the inference rule being used: For all x_5, x_5 will sleep.
 From the fact that for all x_5, x_5 is an astronaut we can infer that for all x_5, x_5 is an astronaut or x_5 will climb a mountain via universal addition. From the fact that for all x_5, x_5 will sleep if and only if x_5 is an astronaut we can infer that for all x_5, if x_5 were an astronaut, then x_5 is sleeping via universal biconditional elimination. Finally, from the fact that for all x_5, if x_5 were an astronaut, then x_5 is sleeping, that for all x_5, if x_5 climbs a mountain, then x_5 is sleeping, and that for all x_5, x_5 is an astronaut or x_5 will climb a mountain we can infer that for all x_5, x_5 will sleep via universal disjunction elimination. Therefore, the answer is yes.