Consider the following premises. For all x_12, x_12 is happy if and only if x_12 will listen to a song. Karen is happy if and only if Karen is listening to a song. Can we infer the following from them? Karen is listening to a song if and only if Karen is happy.
 From the fact that for all x_12, x_12 is happy if and only if x_12 will listen to a song we can infer that Karen is happy if and only if Karen is listening to a song. From the fact that Karen is happy if and only if Karen is listening to a song we can infer that if Karen were happy, then Karen is listening to a song. From the fact that Karen is happy if and only if Karen is listening to a song we can infer that if Karen listens to a song, then Karen is happy. Finally, from the fact that if Karen listens to a song, then Karen is happy, and that if Karen were happy, then Karen is listening to a song we can infer that Karen is listening to a song if and only if Karen is happy. Therefore, the answer is yes.