Consider the following premises. For all y, y will play a game or y will not drink water. There is at least one y for which y will not play a game. For all y, y will drink water or y will not 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 for which y will not drink water.
 From the fact that for all y, y will play a game or y will not drink water we can infer that for all y, if y drinks water, then y is playing a game via universal rewriting. From the fact that for all y, y will drink water or y will not play a game we can infer that for all y, if y plays a game, then y is drinking water via universal rewriting. From the fact that for all y, if y drinks water, then y is playing a game, and that for all y, if y plays a game, then y is drinking water we can infer that for all y, y will drink water if and only if y will play a game via universal biconditional introduction. Finally, from the fact that for all y, y will drink water if and only if y will play a game, and that there is at least one y for which y will not play a game we can infer that there is at least one y for which y will not drink water via existential biconditional elimination. Therefore, the answer is yes.