Consider the following premises. There is at least one z for which z is a doctor if and only if z will listen to a song. For all z, z will listen to a song if and only if z will read a book. For all z, z will not read a book. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z for which z is not a doctor.
 From the fact that for all z, z will listen to a song if and only if z will read a book, and that for all z, z will not read a book we can infer that for all z, z will not listen to a song via universal biconditional elimination. Finally, from the fact that there is at least one z for which z is a doctor if and only if z will listen to a song, and that for all z, z will not listen to a song we can infer that there is at least one z for which z is not a doctor via existential biconditional elimination. Therefore, the answer is yes.