Consider the following premises. Elisabeth is a scientist if and only if Karen is a janitor. Charles is not sleeping. If Karen were bored, then Elisabeth is a scientist. Charles is sleeping if and only if Karen is a janitor. Can we infer the following from them? If we can, name the inference rule being used: Karen is not bored.
 From the fact that Charles is sleeping if and only if Karen is a janitor, and that Charles is not sleeping we can infer that Karen is not a janitor via biconditional elimination. From the fact that Elisabeth is a scientist if and only if Karen is a janitor, and that Karen is not a janitor we can infer that Elisabeth is not a scientist via biconditional elimination. Finally, from the fact that if Karen were bored, then Elisabeth is a scientist, and that Elisabeth is not a scientist we can infer that Karen is not bored via modus tollens. Therefore, the answer is yes.