Consider the following premises. Barbara is an astronaut or it is not late. If Barbara writes a letter, then it is late. Barbara is writing a letter or Barbara is not an astronaut. Can we infer the following from them? If we can, name the inference rule being used: It is late if and only if Barbara is an astronaut.
 From the fact that Barbara is writing a letter or Barbara is not an astronaut we can infer that if Barbara were an astronaut, then Barbara is writing a letter via rewriting. From the fact that Barbara is an astronaut or it is not late we can infer that if it is late, then Barbara is an astronaut via rewriting. From the fact that if Barbara were an astronaut, then Barbara is writing a letter, and that if Barbara writes a letter, then it is late we can infer that if Barbara were an astronaut, then it is late via transitivity. Finally, from the fact that if it is late, then Barbara is an astronaut, and that if Barbara were an astronaut, then it is late we can infer that it is late if and only if Barbara is an astronaut via biconditional introduction. Therefore, the answer is yes.