Consider the following premises. For all x_5, x_5 will play tennis or x_5 will not play a game. There is at least one x_5 for which if x_5 plays tennis, then x_5 is a politician. For all x_5, if x_5 were a politician, 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 play a game if and only if x_5 will play tennis.
 From the fact that there is at least one x_5 for which if x_5 plays tennis, then x_5 is a politician, and that for all x_5, if x_5 were a politician, then x_5 is playing a game we can infer that there is at least one x_5 for which if x_5 plays tennis, then x_5 is playing a game. From the fact that for all x_5, x_5 will play tennis or x_5 will not play a game we can infer that for all x_5, if x_5 plays a game, then x_5 is playing tennis. Finally, from the fact that for all x_5, if x_5 plays a game, then x_5 is playing tennis, and that there is at least one x_5 for which if x_5 plays tennis, then x_5 is playing a game we can infer that there is at least one x_5 for which x_5 will play a game if and only if x_5 will play tennis. Therefore, the answer is yes.