Consider the following premises. For all y_2, y_2 will listen to a song if and only if y_2 will play a game. For all y_2, if y_2 were poor, then y_2 is playing a game. For all y_2, if y_2 plays a game, then y_2 is listening to a song. For all y_2, if y_2 listens to a song, then y_2 is poor. Can we infer the following from them? If we can, name the inference rule being used: For all y_2, y_2 will listen to a song if and only if y_2 will play a game.
That is one of the premises. Therefore, the answer is yes.