Consider the following premises. For all y_4, if y_4 were fast, then y_4 is playing a game. For all y_4, if y_4 plays a game, then y_4 is cooking. For all y_4, if y_4 cooks, then y_4 is an artist. For all y_4, y_4 is fast if and only if y_4 is an artist. 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 is fast 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 cooking, and that for all y_4, if y_4 cooks, then y_4 is an artist we can infer that for all y_4, if y_4 plays a game, then y_4 is an artist via universal transitivity. From the fact that for all y_4, y_4 is fast if and only if y_4 is an artist we can infer that for all y_4, if y_4 were an artist, then y_4 is fast via universal biconditional elimination. From the fact that for all y_4, if y_4 plays a game, then y_4 is an artist, and that for all y_4, if y_4 were an artist, then y_4 is fast we can infer that for all y_4, if y_4 plays a game, then y_4 is fast via universal transitivity. Finally, from the fact that for all y_4, if y_4 were fast, then y_4 is playing a game, and that for all y_4, if y_4 plays a game, then y_4 is fast we can infer that for all y_4, y_4 is fast if and only if y_4 will play a game via universal biconditional introduction, which contradicts that There is at least one y_4 for which if is not true that y_4 is fast if and only if y_4 will play a game. Therefore, the answer is no.