Consider the following premises. If Thomas makes tea, then Linda is cooking. If Linda cooks, then Joseph is a writer. Joseph is not a writer. If it is cloudy, then Thomas is making tea. If Joseph were a writer, then it is cloudy. Can we infer the following from them? If we can, name the inference rule being used: It is not cloudy.
 From the fact that if Thomas makes tea, then Linda is cooking, and that if Linda cooks, then Joseph is a writer we can infer that if Thomas makes tea, then Joseph is a writer via transitivity. From the fact that if it is cloudy, then Thomas is making tea, and that if Thomas makes tea, then Joseph is a writer we can infer that if it is cloudy, then Joseph is a writer via transitivity. From the fact that if it is cloudy, then Joseph is a writer, and that if Joseph were a writer, then it is cloudy we can infer that it is cloudy if and only if Joseph is a writer via biconditional introduction. Finally, from the fact that it is cloudy if and only if Joseph is a writer, and that Joseph is not a writer we can infer that it is not cloudy via biconditional elimination. Therefore, the answer is yes.