Consider the following premises. There is at least one x_7 for which if x_7 listens to a song, then x_7 is writing a letter. For all x_7, if x_7 writes a letter, then x_7 is listening to a song. For all x_7, if x_7 listens to a song, then x_7 is writing a letter. Can we infer the following from them? For all x_7, if is not true that x_7 will listen to a song if and only if x_7 will write a letter.
 From the fact that for all x_7, if x_7 writes a letter, then x_7 is listening to a song, and that there is at least one x_7 for which if x_7 listens to a song, then x_7 is writing a letter we can infer that there is at least one x_7 for which x_7 will write a letter if and only if x_7 will listen to a song. From the fact that there is at least one x_7 for which x_7 will write a letter if and only if x_7 will listen to a song we can infer that there is at least one x_7 for which if x_7 writes a letter, then x_7 is listening to a song. Finally, from the fact that for all x_7, if x_7 listens to a song, then x_7 is writing a letter, and that there is at least one x_7 for which if x_7 writes a letter, then x_7 is listening to a song we can infer that there is at least one x_7 for which x_7 will listen to a song if and only if x_7 will write a letter, which contradicts that For all x_7, if is not true that x_7 will listen to a song if and only if x_7 will write a letter. Therefore, the answer is no.