Consider the following premises. John is bored or Thomas is not a bartender. If Elisabeth were happy, then John is bored. Thomas is a bartender or Elisabeth is happy. It is cold. Can we infer the following from them? If we can, name the inference rule being used: John is not bored.
 From the fact that John is bored or Thomas is not a bartender we can infer that if Thomas were a bartender, then John is bored via rewriting. Finally, from the fact that if Thomas were a bartender, then John is bored, that if Elisabeth were happy, then John is bored, and that Thomas is a bartender or Elisabeth is happy we can infer that John is bored via disjunction elimination, which contradicts that John is not bored. Therefore, the answer is no.