Consider the following premises. For all z, if z were a janitor, then z is writing a letter. For all z, if z were excited, then z is a janitor. For all z, if z writes a letter, then z is excited. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z for which if is not true that z is excited if and only if z will write a letter.
 From the fact that for all z, if z were excited, then z is a janitor, and that for all z, if z were a janitor, then z is writing a letter we can infer that for all z, if z were excited, then z is writing a letter via universal transitivity. Finally, from the fact that for all z, if z were excited, then z is writing a letter, and that for all z, if z writes a letter, then z is excited we can infer that for all z, z is excited if and only if z will write a letter via universal biconditional introduction, which contradicts that There is at least one z for which if is not true that z is excited if and only if z will write a letter. Therefore, the answer is no.