Consider the following premises. For all y, y will climb a mountain or y is not sad. For all y, y will play squash or y will not climb a mountain. For all y, y is sad or y will not play squash. Can we infer the following from them? If we can, name the inference rule being used: For all y, y will play squash if and only if y will climb a mountain.
 From the fact that for all y, y is sad or y will not play squash we can infer that for all y, if y plays squash, then y is sad via universal rewriting. From the fact that for all y, y will climb a mountain or y is not sad we can infer that for all y, if y were sad, then y is climbing a mountain via universal rewriting. From the fact that for all y, y will play squash or y will not climb a mountain we can infer that for all y, if y climbs a mountain, then y is playing squash via universal rewriting. From the fact that for all y, if y plays squash, then y is sad, and that for all y, if y were sad, then y is climbing a mountain we can infer that for all y, if y plays squash, then y is climbing a mountain via universal transitivity. Finally, from the fact that for all y, if y plays squash, then y is climbing a mountain, and that for all y, if y climbs a mountain, then y is playing squash we can infer that for all y, y will play squash if and only if y will climb a mountain via universal biconditional introduction. Therefore, the answer is yes.