Consider the following premises. It is sunny or Barbara is not a mechanic. If it is sunny, then it is late. If it is late, then Michael is a janitor. Barbara is a mechanic or it is overcast. If it is overcast, then Michael is a janitor. Can we infer the following from them? If we can, name the inference rule being used: Michael is a janitor.
 From the fact that it is sunny or Barbara is not a mechanic we can infer that if Barbara were a mechanic, then it is sunny via rewriting. From the fact that if Barbara were a mechanic, then it is sunny, and that if it is sunny, then it is late we can infer that if Barbara were a mechanic, then it is late via transitivity. From the fact that if Barbara were a mechanic, then it is late, and that if it is late, then Michael is a janitor we can infer that if Barbara were a mechanic, then Michael is a janitor via transitivity. Finally, from the fact that if Barbara were a mechanic, then Michael is a janitor, that if it is overcast, then Michael is a janitor, and that Barbara is a mechanic or it is overcast we can infer that Michael is a janitor via disjunction elimination. Therefore, the answer is yes.