Consider the following premises. There is at least one x_15 for which if x_15 were a mechanic, then x_15 is a janitor. For all x_15, x_15 is a mechanic or x_15 is not a nurse. For all x_15, if x_15 were a police officer, then x_15 is a nurse. For all x_15, x_15 is a police officer if and only if x_15 is a janitor. Can we infer the following from them? There is at least one x_15 for which x_15 is a janitor if and only if x_15 is a nurse.
 From the fact that for all x_15, x_15 is a mechanic or x_15 is not a nurse we can infer that for all x_15, if x_15 were a nurse, then x_15 is a mechanic. From the fact that for all x_15, x_15 is a police officer if and only if x_15 is a janitor we can infer that for all x_15, if x_15 were a janitor, then x_15 is a police officer. From the fact that for all x_15, if x_15 were a janitor, then x_15 is a police officer, and that for all x_15, if x_15 were a police officer, then x_15 is a nurse we can infer that for all x_15, if x_15 were a janitor, then x_15 is a nurse. From the fact that for all x_15, if x_15 were a nurse, then x_15 is a mechanic, and that there is at least one x_15 for which if x_15 were a mechanic, then x_15 is a janitor we can infer that there is at least one x_15 for which if x_15 were a nurse, then x_15 is a janitor. Finally, from the fact that for all x_15, if x_15 were a janitor, then x_15 is a nurse, and that there is at least one x_15 for which if x_15 were a nurse, then x_15 is a janitor we can infer that there is at least one x_15 for which x_15 is a janitor if and only if x_15 is a nurse. Therefore, the answer is yes.