Consider the following premises. For all x, x is fast or x will not listen to a song. For all x, if x were fast, then x is cooking. There is at least one x for which if x cooks, then x is listening to a song. Can we infer the following from them? There is at least one x for which x is fast if and only if x will listen to a song.
 From the fact that for all x, if x were fast, then x is cooking, and that there is at least one x for which if x cooks, then x is listening to a song we can infer that there is at least one x for which if x were fast, then x is listening to a song. From the fact that for all x, x is fast or x will not listen to a song we can infer that for all x, if x listens to a song, then x is fast. Finally, from the fact that there is at least one x for which if x were fast, then x is listening to a song, and that for all x, if x listens to a song, then x is fast we can infer that there is at least one x for which x is fast if and only if x will listen to a song. Therefore, the answer is yes.