Consider the following premises. If Joseph were a writer, then Barbara is cooking. For all x_12, if x_12 were a writer, then x_12 is rich. If Joseph were rich, then Joseph is a writer. Can we infer the following from them? If Joseph were rich, then Barbara is cooking.
 From the fact that for all x_12, if x_12 were a writer, then x_12 is rich we can infer that if Joseph were a writer, then Joseph is rich. From the fact that if Joseph were a writer, then Joseph is rich, and that if Joseph were rich, then Joseph is a writer we can infer that Joseph is a writer if and only if Joseph is rich. From the fact that Joseph is a writer if and only if Joseph is rich we can infer that if Joseph were rich, then Joseph is a writer. Finally, from the fact that if Joseph were rich, then Joseph is a writer, and that if Joseph were a writer, then Barbara is cooking we can infer that if Joseph were rich, then Barbara is cooking. Therefore, the answer is yes.