Consider the following premises. For all y, if y cooks, then y is curious. For all y, y will play a game if and only if y is curious. There is at least one y for which if y plays a game, then y is cooking. 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 play a game if and only if y is curious.
 From the fact that there is at least one y for which if y plays a game, then y is cooking, and that for all y, if y cooks, then y is curious we can infer that there is at least one y for which if y plays a game, then y is curious via existential transitivity. From the fact that for all y, y will play a game if and only if y is curious we can infer that for all y, if y were curious, then y is playing a game via universal biconditional elimination. Finally, from the fact that there is at least one y for which if y plays a game, then y is curious, and that for all y, if y were curious, then y is playing a game we can infer that there is at least one y for which y will play a game if and only if y is curious via existential biconditional introduction. Therefore, the answer is yes.