Consider the following premises. It is sunny if and only if Barbara is cooking. If Barbara cooks, then Joseph is a mechanic. If Joseph were a mechanic, then Barbara is cooking. Joseph is not a mechanic. Can we infer the following from them? It is not sunny.
 From the fact that if Joseph were a mechanic, then Barbara is cooking, and that if Barbara cooks, then Joseph is a mechanic we can infer that Joseph is a mechanic if and only if Barbara is cooking. From the fact that Joseph is a mechanic if and only if Barbara is cooking, and that Joseph is not a mechanic we can infer that Barbara is not cooking. Finally, from the fact that it is sunny if and only if Barbara is cooking, and that Barbara is not cooking we can infer that it is not sunny. Therefore, the answer is yes.