Consider the following premises. For all y_4, y_4 will go running if and only if y_4 will play a game. For all y_4, y_4 will go running if and only if y_4 will play a game. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y_4 for which if is not true that y_4 will play a game if and only if y_4 will go running.
 From the fact that for all y_4, y_4 will go running if and only if y_4 will play a game we can infer that for all y_4, if y_4 plays a game, then y_4 is running via universal biconditional elimination. From the fact that for all y_4, y_4 will go running if and only if y_4 will play a game we can infer that for all y_4, if y_4 goes running, then y_4 is playing a game via universal biconditional elimination. Finally, from the fact that for all y_4, if y_4 plays a game, then y_4 is running, and that for all y_4, if y_4 goes running, then y_4 is playing a game we can infer that for all y_4, y_4 will play a game if and only if y_4 will go running via universal biconditional introduction, which contradicts that There is at least one y_4 for which if is not true that y_4 will play a game if and only if y_4 will go running. Therefore, the answer is no.