Consider the following premises. For all x_7, x_7 will listen to a song or x_7 will not write a letter. For all x_7, x_7 will write a letter or x_7 will not listen to a song. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_7 for which the fact that x_7 listens to a song does not imply that x_7 is writing a letter.
 From the fact that for all x_7, x_7 will listen to a song or x_7 will not write a letter we can infer that for all x_7, if x_7 writes a letter, then x_7 is listening to a song via universal rewriting. From the fact that for all x_7, x_7 will write a letter or x_7 will not listen to a song we can infer that for all x_7, if x_7 listens to a song, then x_7 is writing a letter via universal rewriting. From the fact that for all x_7, if x_7 listens to a song, then x_7 is writing a letter, and that for all x_7, if x_7 writes a letter, then x_7 is listening to a song we can infer that for all x_7, x_7 will listen to a song if and only if x_7 will write a letter via universal biconditional introduction. Finally, from the fact that for all x_7, x_7 will listen to a song if and only if x_7 will write a letter we can infer that for all x_7, if x_7 listens to a song, then x_7 is writing a letter via universal biconditional elimination, which contradicts that There is at least one x_7 for which the fact that x_7 listens to a song does not imply that x_7 is writing a letter. Therefore, the answer is no.