Consider the following premises. For all x_5, if x_5 were a nurse, then x_5 is reading a book. For all x_5, if x_5 were a mechanic, then x_5 is a nurse. For all x_5, if x_5 were a police officer, then x_5 is a mechanic. There is at least one x_5 for which if x_5 reads a book, then x_5 is a police officer. Can we infer the following from them? There is at least one x_5 for which x_5 is a police officer if and only if x_5 will read a book.
 From the fact that for all x_5, if x_5 were a mechanic, then x_5 is a nurse, and that for all x_5, if x_5 were a nurse, then x_5 is reading a book we can infer that for all x_5, if x_5 were a mechanic, then x_5 is reading a book. From the fact that for all x_5, if x_5 were a police officer, then x_5 is a mechanic, and that for all x_5, if x_5 were a mechanic, then x_5 is reading a book we can infer that for all x_5, if x_5 were a police officer, then x_5 is reading a book. Finally, from the fact that for all x_5, if x_5 were a police officer, then x_5 is reading a book, and that there is at least one x_5 for which if x_5 reads a book, then x_5 is a police officer we can infer that there is at least one x_5 for which x_5 is a police officer if and only if x_5 will read a book. Therefore, the answer is yes.