Consider the following premises. For all z, z is not an office clerk. For all z, z is an office clerk if and only if z is an astronaut. There is at least one z for which z is an astronaut if and only if z will listen to a song. Can we infer the following from them? If we can, name the inference rule being used: For all z, z will listen to a song.
 From the fact that for all z, z is an office clerk if and only if z is an astronaut, and that for all z, z is not an office clerk we can infer that for all z, z is not an astronaut via universal biconditional elimination. From the fact that there is at least one z for which z is an astronaut if and only if z will listen to a song we can infer that there is at least one z for which if z listens to a song, then z is an astronaut via existential biconditional elimination. Finally, from the fact that there is at least one z for which if z listens to a song, then z is an astronaut, and that for all z, z is not an astronaut we can infer that there is at least one z for which z will not listen to a song via existential modus tollens, which contradicts that For all z, z will listen to a song. Therefore, the answer is no.