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