Consider the following premises. For all x_6, x_6 is happy if and only if x_6 will listen to a song. For all x_6, x_6 is happy or x_6 will not listen to a song. Can we infer the following from them? For all x_6, x_6 will listen to a song if and only if x_6 is happy.
 From the fact that for all x_6, x_6 is happy or x_6 will not listen to a song we can infer that for all x_6, if x_6 listens to a song, then x_6 is happy. From the fact that for all x_6, x_6 is happy if and only if x_6 will listen to a song we can infer that for all x_6, if x_6 were happy, then x_6 is listening to a song. Finally, from the fact that for all x_6, if x_6 listens to a song, then x_6 is happy, and that for all x_6, if x_6 were happy, then x_6 is listening to a song we can infer that for all x_6, x_6 will listen to a song if and only if x_6 is happy. Therefore, the answer is yes.