Consider the following premises. There is at least one x for which if x were a janitor, then x is a nurse. For all x, if x were a nurse, then x is a janitor. For all x, if x were a janitor, then x is a nurse. Can we infer the following from them? There is at least one x for which x is a janitor if and only if x is a nurse.
 From the fact that for all x, if x were a janitor, then x is a nurse, and that for all x, if x were a nurse, then x is a janitor we can infer that for all x, x is a janitor if and only if x is a nurse. From the fact that for all x, x is a janitor if and only if x is a nurse we can infer that for all x, if x were a nurse, then x is a janitor. Finally, from the fact that there is at least one x for which if x were a janitor, then x is a nurse, and that for all x, if x were a nurse, then x is a janitor we can infer that there is at least one x for which x is a janitor if and only if x is a nurse. Therefore, the answer is yes.