Consider the following premises. If Mary writes a letter, then Barbara is a doctor. If it is cold, then Thomas is a carpenter. Thomas is a carpenter if and only if Mary is writing a letter. It is cold. Can we infer the following from them? If we can, name the inference rule being used: Barbara is not a doctor.
 From the fact that Thomas is a carpenter if and only if Mary is writing a letter we can infer that if Thomas were a carpenter, then Mary is writing a letter via biconditional elimination. From the fact that if it is cold, then Thomas is a carpenter, and that if Thomas were a carpenter, then Mary is writing a letter we can infer that if it is cold, then Mary is writing a letter via transitivity. From the fact that if it is cold, then Mary is writing a letter, and that if Mary writes a letter, then Barbara is a doctor we can infer that if it is cold, then Barbara is a doctor via transitivity. Finally, from the fact that if it is cold, then Barbara is a doctor, and that it is cold we can infer that Barbara is a doctor via modus ponens, which contradicts that Barbara is not a doctor. Therefore, the answer is no.