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