Consider the following premises. If Robert writes a letter, then Karen is happy. If Karen were happy, then it is raining. Karen is not happy. If it rains, then Robert is writing a letter. Can we infer the following from them? If we can, name the inference rule being used: It is not raining.
 From the fact that if it rains, then Robert is writing a letter, and that if Robert writes a letter, then Karen is happy we can infer that if it rains, then Karen is happy via transitivity. From the fact that if Karen were happy, then it is raining, and that if it rains, then Karen is happy we can infer that Karen is happy if and only if it is raining via biconditional introduction. Finally, from the fact that Karen is happy if and only if it is raining, and that Karen is not happy we can infer that it is not raining via biconditional elimination. Therefore, the answer is yes.