Consider the following premises. Charles is a politician if and only if Joseph is poor. Charles is a politician if and only if it is sunny. If it is sunny, then it is late. It is not late. Can we infer the following from them? If we can, name the inference rule being used: Joseph is not poor.
 From the fact that Charles is a politician if and only if it is sunny we can infer that if Charles were a politician, then it is sunny via biconditional elimination. From the fact that if Charles were a politician, then it is sunny, and that if it is sunny, then it is late we can infer that if Charles were a politician, then it is late via transitivity. From the fact that if Charles were a politician, then it is late, and that it is not late we can infer that Charles is not a politician via modus tollens. From the fact that Charles is a politician if and only if Joseph is poor we can infer that if Joseph were poor, then Charles is a politician via biconditional elimination. Finally, from the fact that if Joseph were poor, then Charles is a politician, and that Charles is not a politician we can infer that Joseph is not poor via modus tollens. Therefore, the answer is yes.