Consider the following premises. If Sarah sleeps, then Sarah is a doctor. For all x_8, x_8 is a doctor if and only if x_8 will sleep. Can we infer the following from them? Sarah is a doctor if and only if Sarah is sleeping.
 From the fact that for all x_8, x_8 is a doctor if and only if x_8 will sleep we can infer that for all x_8, if x_8 were a doctor, then x_8 is sleeping. From the fact that for all x_8, if x_8 were a doctor, then x_8 is sleeping we can infer that if Sarah were a doctor, then Sarah is sleeping. Finally, from the fact that if Sarah were a doctor, then Sarah is sleeping, and that if Sarah sleeps, then Sarah is a doctor we can infer that Sarah is a doctor if and only if Sarah is sleeping. Therefore, the answer is yes.