Consider the following premises. Barbara is not an astronaut. If it rains, then Susan is poor. If Susan were poor, then it is raining. It is raining if and only if Barbara is an astronaut. If Barbara were an astronaut, then it is raining. Can we infer the following from them? If we can, name the inference rule being used: Susan is not poor.
 From the fact that it is raining if and only if Barbara is an astronaut we can infer that if it rains, then Barbara is an astronaut via biconditional elimination. From the fact that if it rains, then Barbara is an astronaut, and that if Barbara were an astronaut, then it is raining we can infer that it is raining if and only if Barbara is an astronaut via biconditional introduction. From the fact that it is raining if and only if Barbara is an astronaut, and that Barbara is not an astronaut we can infer that it is not raining via biconditional elimination. From the fact that if it rains, then Susan is poor, and that if Susan were poor, then it is raining we can infer that it is raining if and only if Susan is poor via biconditional introduction. Finally, from the fact that it is raining if and only if Susan is poor, and that it is not raining we can infer that Susan is not poor via biconditional elimination. Therefore, the answer is yes.