Consider the following premises. For all y_4, if y_4 plays tennis, then y_4 is a taxi driver. For all y_4, if y_4 plays a game, then y_4 is playing tennis. There is at least one y_4 for which y_4 will play a game or y_4 is not a taxi driver. Can we infer the following from them? There is at least one y_4 for which y_4 is a taxi driver if and only if y_4 will play a game.
 From the fact that for all y_4, if y_4 plays a game, then y_4 is playing tennis, and that for all y_4, if y_4 plays tennis, then y_4 is a taxi driver we can infer that for all y_4, if y_4 plays a game, then y_4 is a taxi driver. From the fact that there is at least one y_4 for which y_4 will play a game or y_4 is not a taxi driver we can infer that there is at least one y_4 for which if y_4 were a taxi driver, then y_4 is playing a game. Finally, from the fact that there is at least one y_4 for which if y_4 were a taxi driver, then y_4 is playing a game, and that for all y_4, if y_4 plays a game, then y_4 is a taxi driver we can infer that there is at least one y_4 for which y_4 is a taxi driver if and only if y_4 will play a game. Therefore, the answer is yes.