Consider the following premises. Richard is writing a letter or it is not sunny. It is sunny if and only if Richard is writing a letter. Can we infer the following from them? Richard is writing a letter if and only if it is sunny.
 From the fact that it is sunny if and only if Richard is writing a letter we can infer that if Richard writes a letter, then it is sunny. From the fact that Richard is writing a letter or it is not sunny we can infer that if it is sunny, then Richard is writing a letter. Finally, from the fact that if Richard writes a letter, then it is sunny, and that if it is sunny, then Richard is writing a letter we can infer that Richard is writing a letter if and only if it is sunny. Therefore, the answer is yes.