Consider the following premises. If Karen were happy, then Elisabeth is a doctor. If Elisabeth were a doctor, then Karen is happy. John is poor if and only if Karen is happy. John is not poor. Can we infer the following from them? If we can, name the inference rule being used: Elisabeth is not a doctor.
 From the fact that if Karen were happy, then Elisabeth is a doctor, and that if Elisabeth were a doctor, then Karen is happy we can infer that Karen is happy if and only if Elisabeth is a doctor via biconditional introduction. From the fact that John is poor if and only if Karen is happy, and that John is not poor we can infer that Karen is not happy via biconditional elimination. Finally, from the fact that Karen is happy if and only if Elisabeth is a doctor, and that Karen is not happy we can infer that Elisabeth is not a doctor via biconditional elimination. Therefore, the answer is yes.