Consider the following premises. For all x, x is a soldier or x will not write a letter. There is at least one x for which x will write a letter or x is not a soldier. Can we infer the following from them? There is at least one x for which x will write a letter if and only if x is a soldier.
 From the fact that there is at least one x for which x will write a letter or x is not a soldier we can infer that there is at least one x for which if x were a soldier, then x is writing a letter. From the fact that for all x, x is a soldier or x will not write a letter we can infer that for all x, if x writes a letter, then x is a soldier. Finally, from the fact that for all x, if x writes a letter, then x is a soldier, and that there is at least one x for which if x were a soldier, then x is writing a letter we can infer that there is at least one x for which x will write a letter if and only if x is a soldier. Therefore, the answer is yes.