Consider the following premises. For all x_4, x_4 will play a game or x_4 will not listen to a song. For all x_4, x_4 will not listen to a song. There is at least one x_4 for which if x_4 listens to a song, then x_4 is playing a game. For all x_4, if x_4 plays a game, then x_4 is listening to a song. Can we infer the following from them? If we can, name the inference rule being used: For all x_4, x_4 will play a game.
 From the fact that for all x_4, x_4 will play a game or x_4 will not listen to a song we can infer that for all x_4, if x_4 listens to a song, then x_4 is playing a game via universal rewriting. From the fact that for all x_4, if x_4 listens to a song, then x_4 is playing a game, and that for all x_4, if x_4 plays a game, then x_4 is listening to a song we can infer that for all x_4, x_4 will listen to a song if and only if x_4 will play a game via universal biconditional introduction. From the fact that for all x_4, x_4 will listen to a song if and only if x_4 will play a game we can infer that for all x_4, if x_4 plays a game, then x_4 is listening to a song via universal biconditional elimination. From the fact that for all x_4, if x_4 plays a game, then x_4 is listening to a song, and that there is at least one x_4 for which if x_4 listens to a song, then x_4 is playing a game we can infer that there is at least one x_4 for which x_4 will play a game if and only if x_4 will listen to a song via existential biconditional introduction. Finally, from the fact that there is at least one x_4 for which x_4 will play a game if and only if x_4 will listen to a song, and that for all x_4, x_4 will not listen to a song we can infer that there is at least one x_4 for which x_4 will not play a game via existential biconditional elimination, which contradicts that For all x_4, x_4 will play a game. Therefore, the answer is no.