Consider the following premises. If Linda sleeps, then James is climbing a mountain. James is climbing a mountain or Thomas is not a doctor. Linda is sleeping. Can we infer the following from them? If we can, name the inference rule being used: James is not climbing a mountain.
 From the fact that James is climbing a mountain or Thomas is not a doctor we can infer that if Thomas were a doctor, then James is climbing a mountain via rewriting. From the fact that Linda is sleeping we can infer that Linda is sleeping or Thomas is a doctor via addition. Finally, from the fact that if Linda sleeps, then James is climbing a mountain, that if Thomas were a doctor, then James is climbing a mountain, and that Linda is sleeping or Thomas is a doctor we can infer that James is climbing a mountain via disjunction elimination, which contradicts that James is not climbing a mountain. Therefore, the answer is no.