Consider the following premises. Joseph is running or Patricia is an astronaut. James is writing a letter or Joseph is not running. James is writing a letter if and only if Patricia is an astronaut. Can we infer the following from them? If we can, name the inference rule being used: James is writing a letter.
 From the fact that James is writing a letter or Joseph is not running we can infer that if Joseph goes running, then James is writing a letter via rewriting. From the fact that James is writing a letter if and only if Patricia is an astronaut we can infer that if Patricia were an astronaut, then James is writing a letter via biconditional elimination. Finally, from the fact that if Joseph goes running, then James is writing a letter, that if Patricia were an astronaut, then James is writing a letter, and that Joseph is running or Patricia is an astronaut we can infer that James is writing a letter via disjunction elimination. Therefore, the answer is yes.