Consider the following premises. There is at least one y_1 for which y_1 is a writer or y_1 will play a game. For all y_1, y_1 is a writer if and only if y_1 will listen to a song. For all y_1, y_1 will play a game if and only if y_1 will listen to a song. Can we infer the following from them? If we can, name the inference rule being used: For all y_1, y_1 will not listen to a song.
 From the fact that for all y_1, y_1 is a writer if and only if y_1 will listen to a song we can infer that for all y_1, if y_1 were a writer, then y_1 is listening to a song via universal biconditional elimination. From the fact that for all y_1, y_1 will play a game if and only if y_1 will listen to a song we can infer that for all y_1, if y_1 plays a game, then y_1 is listening to a song via universal biconditional elimination. Finally, from the fact that for all y_1, if y_1 were a writer, then y_1 is listening to a song, that for all y_1, if y_1 plays a game, then y_1 is listening to a song, and that there is at least one y_1 for which y_1 is a writer or y_1 will play a game we can infer that there is at least one y_1 for which y_1 will listen to a song via existential disjunction elimination, which contradicts that For all y_1, y_1 will not listen to a song. Therefore, the answer is no.