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