Consider the following premises. For all x_10, if x_10 were an astronaut, then x_10 is sleeping. For all x_10, if x_10 sleeps, then x_10 is an astronaut. There is at least one x_10 for which x_10 will sleep if and only if x_10 is an astronaut. Can we infer the following from them? There is at least one x_10 for which x_10 is an astronaut if and only if x_10 will sleep.
 From the fact that for all x_10, if x_10 were an astronaut, then x_10 is sleeping, and that for all x_10, if x_10 sleeps, then x_10 is an astronaut we can infer that for all x_10, x_10 is an astronaut if and only if x_10 will sleep. From the fact that there is at least one x_10 for which x_10 will sleep if and only if x_10 is an astronaut we can infer that there is at least one x_10 for which if x_10 sleeps, then x_10 is an astronaut. From the fact that for all x_10, x_10 is an astronaut if and only if x_10 will sleep we can infer that for all x_10, if x_10 were an astronaut, then x_10 is sleeping. Finally, from the fact that for all x_10, if x_10 were an astronaut, then x_10 is sleeping, and that there is at least one x_10 for which if x_10 sleeps, then x_10 is an astronaut we can infer that there is at least one x_10 for which x_10 is an astronaut if and only if x_10 will sleep. Therefore, the answer is yes.