Consider the following premises. For all x, x will write a letter if and only if x will climb a mountain. For all x, x will write a letter if and only if x will climb a mountain. Can we infer the following from them? For all x, x will climb a mountain if and only if x will write a letter.
 From the fact that for all x, x will write a letter if and only if x will climb a mountain we can infer that for all x, if x writes a letter, then x is climbing a mountain. From the fact that for all x, x will write a letter if and only if x will climb a mountain we can infer that for all x, if x climbs a mountain, then x is writing a letter. Finally, from the fact that for all x, if x climbs a mountain, then x is writing a letter, and that for all x, if x writes a letter, then x is climbing a mountain we can infer that for all x, x will climb a mountain if and only if x will write a letter. Therefore, the answer is yes.