Consider the following premises. If James were a mechanic, then Jessica is rich. Jessica is rich or Joseph is not cooking. If it rains, then Jessica is not rich. James is a mechanic. Can we infer the following from them? If we can, name the inference rule being used: It is raining.
 From the fact that James is a mechanic we can infer that James is a mechanic or Joseph is cooking via addition. From the fact that Jessica is rich or Joseph is not cooking we can infer that if Joseph cooks, then Jessica is rich via rewriting. From the fact that if James were a mechanic, then Jessica is rich, that if Joseph cooks, then Jessica is rich, and that James is a mechanic or Joseph is cooking we can infer that Jessica is rich via disjunction elimination. Finally, from the fact that if it rains, then Jessica is not rich, and that Jessica is rich we can infer that it is not raining via modus tollens, which contradicts that It is raining. Therefore, the answer is no.