Consider the following premises. For all z, if z works, then z is listening to a song. There is at least one z for which z will listen to a song if and only if z will work. Can we infer the following from them? For all z, if is not true that z will work if and only if z will listen to a song.
 From the fact that there is at least one z for which z will listen to a song if and only if z will work we can infer that there is at least one z for which if z listens to a song, then z is working. Finally, from the fact that for all z, if z works, then z is listening to a song, and that there is at least one z for which if z listens to a song, then z is working we can infer that there is at least one z for which z will work if and only if z will listen to a song, which contradicts that For all z, if is not true that z will work if and only if z will listen to a song. Therefore, the answer is no.