Consider the following premises. It is sunny if and only if Thomas is rich. Karen is climbing a mountain if and only if it is sunny. Thomas is not rich. Can we infer the following from them? Karen is not climbing a mountain.
 From the fact that it is sunny if and only if Thomas is rich, and that Thomas is not rich we can infer that it is not sunny. Finally, from the fact that Karen is climbing a mountain if and only if it is sunny, and that it is not sunny we can infer that Karen is not climbing a mountain. Therefore, the answer is yes.