Consider the following premises. For all x_5, x_5 is a cashier if and only if x_5 will sleep. For all x_5, x_5 is an astronaut. There is at least one x_5 for which if x_5 were an astronaut, then x_5 is sleeping. Can we infer the following from them? There is at least one x_5 for which x_5 will sleep.
 From the fact that for all x_5, x_5 is a cashier if and only if x_5 will sleep we can infer that for all x_5, if x_5 were a cashier, then x_5 is sleeping. 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 is a cashier. Finally, from the fact that there is at least one x_5 for which if x_5 were an astronaut, then x_5 is sleeping, that for all x_5, if x_5 were a cashier, then x_5 is sleeping, and that for all x_5, x_5 is an astronaut or x_5 is a cashier we can infer that there is at least one x_5 for which x_5 will sleep. Therefore, the answer is yes.