Consider the following premises. For all y_1, if y_1 goes running, then y_1 is playing a game. For all y_1, if y_1 plays a game, then y_1 is running. There is at least one y_1 for which y_1 will go running if and only if y_1 will play a game. Can we infer the following from them? If we can, name the inference rule being used: For all y_1, if is not true that y_1 will play a game if and only if y_1 will go running.
That contradicts the premises. Therefore the answer is no.