Consider the following premises. For all x_2, if x_2 plays squash, then x_2 is listening to a song. For all x_2, if x_2 cooks, then x_2 is listening to a song. For all x_2, x_2 will write a letter or x_2 will play squash. For all x_2, x_2 will not write a letter or x_2 will cook. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_2 for which x_2 will not listen to a song.
 From the fact that for all x_2, x_2 will write a letter or x_2 will play squash, and that for all x_2, x_2 will not write a letter or x_2 will cook we can infer that for all x_2, x_2 will play squash or x_2 will cook via universal resolution. Finally, from the fact that for all x_2, if x_2 plays squash, then x_2 is listening to a song, that for all x_2, if x_2 cooks, then x_2 is listening to a song, and that for all x_2, x_2 will play squash or x_2 will cook we can infer that for all x_2, x_2 will listen to a song via universal disjunction elimination, which contradicts that There is at least one x_2 for which x_2 will not listen to a song. Therefore, the answer is no.