Consider the following premises. It is raining if and only if John is writing a letter. Robert is sleeping or John is writing a letter. Robert is sleeping if and only if it is raining. Can we infer the following from them? It is raining.
 From the fact that Robert is sleeping if and only if it is raining we can infer that if Robert sleeps, then it is raining. From the fact that it is raining if and only if John is writing a letter we can infer that if John writes a letter, then it is raining. Finally, from the fact that if Robert sleeps, then it is raining, that if John writes a letter, then it is raining, and that Robert is sleeping or John is writing a letter we can infer that it is raining. Therefore, the answer is yes.