Consider the following premises. There is at least one x_5 for which if x_5 works, then x_5 is playing a game. For all x_5, if x_5 plays a game, then x_5 is working. For all x_5, if x_5 works, then x_5 is playing a game. Can we infer the following from them? There is at least one x_5 for which x_5 will work if and only if x_5 will play a game.
 From the fact that for all x_5, if x_5 plays a game, then x_5 is working, and that for all x_5, if x_5 works, then x_5 is playing a game we can infer that for all x_5, x_5 will play a game if and only if x_5 will work. From the fact that for all x_5, x_5 will play a game if and only if x_5 will work we can infer that for all x_5, if x_5 plays a game, then x_5 is working. Finally, from the fact that there is at least one x_5 for which if x_5 works, then x_5 is playing a game, and that for all x_5, if x_5 plays a game, then x_5 is working we can infer that there is at least one x_5 for which x_5 will work if and only if x_5 will play a game. Therefore, the answer is yes.