Consider the following premises. Karen is a carpenter or Barbara is not a poet. If Sarah were a mechanic, then Barbara is a poet. Sarah is a mechanic or Karen is not a carpenter. Can we infer the following from them? If we can, name the inference rule being used: If is not true that Barbara is a poet if and only if Karen is a carpenter.
 From the fact that Sarah is a mechanic or Karen is not a carpenter we can infer that if Karen were a carpenter, then Sarah is a mechanic via rewriting. From the fact that Karen is a carpenter or Barbara is not a poet we can infer that if Barbara were a poet, then Karen is a carpenter via rewriting. From the fact that if Karen were a carpenter, then Sarah is a mechanic, and that if Sarah were a mechanic, then Barbara is a poet we can infer that if Karen were a carpenter, then Barbara is a poet via transitivity. Finally, from the fact that if Barbara were a poet, then Karen is a carpenter, and that if Karen were a carpenter, then Barbara is a poet we can infer that Barbara is a poet if and only if Karen is a carpenter via biconditional introduction, which contradicts that If is not true that Barbara is a poet if and only if Karen is a carpenter. Therefore, the answer is no.