Consider the following premises. If Barbara were excited, then it is overcast. John is cooking if and only if Joseph is an electrician. If John cooks, then Barbara is excited. Can we infer the following from them? If Joseph were an electrician, then it is overcast.
 From the fact that if John cooks, then Barbara is excited, and that if Barbara were excited, then it is overcast we can infer that if John cooks, then it is overcast. From the fact that John is cooking if and only if Joseph is an electrician we can infer that if Joseph were an electrician, then John is cooking. Finally, from the fact that if Joseph were an electrician, then John is cooking, and that if John cooks, then it is overcast we can infer that if Joseph were an electrician, then it is overcast. Therefore, the answer is yes.