Consider the following premises. If Joseph were happy, then John is reading a book. It is overcast or John is not reading a book. If it is overcast, then Joseph is happy. Can we infer the following from them? If is not true that it is overcast if and only if John is reading a book.
 From the fact that if it is overcast, then Joseph is happy, and that if Joseph were happy, then John is reading a book we can infer that if it is overcast, then John is reading a book. From the fact that it is overcast or John is not reading a book we can infer that if John reads a book, then it is overcast. Finally, from the fact that if it is overcast, then John is reading a book, and that if John reads a book, then it is overcast we can infer that it is overcast if and only if John is reading a book, which contradicts that If is not true that it is overcast if and only if John is reading a book. Therefore, the answer is no.