Consider the following premises. For all x_9, x_9 is a politician if and only if x_9 is an astronaut. If Richard were a politician, then Richard is an astronaut. Can we infer the following from them? If we can, name the inference rule being used: Richard is a politician if and only if Richard is an astronaut.
 From the fact that for all x_9, x_9 is a politician if and only if x_9 is an astronaut we can infer that for all x_9, if x_9 were an astronaut, then x_9 is a politician via universal biconditional elimination. From the fact that for all x_9, if x_9 were an astronaut, then x_9 is a politician we can infer that if Richard were an astronaut, then Richard is a politician via universal instantiation. Finally, from the fact that if Richard were a politician, then Richard is an astronaut, and that if Richard were an astronaut, then Richard is a politician we can infer that Richard is a politician if and only if Richard is an astronaut via biconditional introduction. Therefore, the answer is yes.