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