Consider the following premises. Patricia is reading a book if and only if it is cloudy. Patricia is not reading a book. If Robert were an astronaut, then Patricia is reading a book. If it is cloudy, then Robert is an astronaut. Can we infer the following from them? If we can, name the inference rule being used: Robert is not an astronaut.
 From the fact that Patricia is reading a book if and only if it is cloudy we can infer that if Patricia reads a book, then it is cloudy via biconditional elimination. From the fact that if Patricia reads a book, then it is cloudy, and that if it is cloudy, then Robert is an astronaut we can infer that if Patricia reads a book, then Robert is an astronaut via transitivity. From the fact that if Robert were an astronaut, then Patricia is reading a book, and that if Patricia reads a book, then Robert is an astronaut we can infer that Robert is an astronaut if and only if Patricia is reading a book via biconditional introduction. Finally, from the fact that Robert is an astronaut if and only if Patricia is reading a book, and that Patricia is not reading a book we can infer that Robert is not an astronaut via biconditional elimination. Therefore, the answer is yes.