Consider the following premises. For all x_1, x_1 is a doctor if and only if x_1 is a nurse. For all x_1, x_1 is a doctor if and only if x_1 is a nurse. Can we infer the following from them? For all x_1, x_1 is a nurse if and only if x_1 is a doctor.
 From the fact that for all x_1, x_1 is a doctor if and only if x_1 is a nurse we can infer that for all x_1, if x_1 were a nurse, then x_1 is a doctor. From the fact that for all x_1, x_1 is a doctor if and only if x_1 is a nurse we can infer that for all x_1, if x_1 were a doctor, then x_1 is a nurse. Finally, from the fact that for all x_1, if x_1 were a nurse, then x_1 is a doctor, and that for all x_1, if x_1 were a doctor, then x_1 is a nurse we can infer that for all x_1, x_1 is a nurse if and only if x_1 is a doctor. Therefore, the answer is yes.