Consider the following premises. There is at least one y for which y is not fast. For all y, if y were poor, then y is playing a game. For all y, y is fast if and only if y will play a game. Can we infer the following from them? If we can, name the inference rule being used: For all y, y is poor.
 From the fact that for all y, y is fast if and only if y will play a game we can infer that for all y, if y plays a game, then y is fast via universal biconditional elimination. From the fact that for all y, if y were poor, then y is playing a game, and that for all y, if y plays a game, then y is fast we can infer that for all y, if y were poor, then y is fast via universal transitivity. Finally, from the fact that for all y, if y were poor, then y is fast, and that there is at least one y for which y is not fast we can infer that there is at least one y for which y is not poor via existential modus tollens, which contradicts that For all y, y is poor. Therefore, the answer is no.