Consider the following premises. If Charles were a nurse, then Elisabeth is a mechanic. If Elisabeth were a mechanic, then Charles is a nurse. Elisabeth is a mechanic or Barbara is not playing a game. Can we infer the following from them? If we can, name the inference rule being used: If Barbara plays a game, then Charles is a nurse.
 From the fact that if Charles were a nurse, then Elisabeth is a mechanic, and that if Elisabeth were a mechanic, then Charles is a nurse we can infer that Charles is a nurse if and only if Elisabeth is a mechanic via biconditional introduction. From the fact that Charles is a nurse if and only if Elisabeth is a mechanic we can infer that if Elisabeth were a mechanic, then Charles is a nurse via biconditional elimination. From the fact that Elisabeth is a mechanic or Barbara is not playing a game we can infer that if Barbara plays a game, then Elisabeth is a mechanic via rewriting. Finally, from the fact that if Barbara plays a game, then Elisabeth is a mechanic, and that if Elisabeth were a mechanic, then Charles is a nurse we can infer that if Barbara plays a game, then Charles is a nurse via transitivity. Therefore, the answer is yes.