Consider the following premises. If Elisabeth cooks, then Barbara is playing a game. If it is cloudy, then Elisabeth is cooking. Elisabeth is cooking if and only if it is cloudy. It is not cloudy. If Barbara plays a game, then Elisabeth is cooking. Can we infer the following from them? Barbara is not playing a game.
 From the fact that Elisabeth is cooking if and only if it is cloudy we can infer that if Elisabeth cooks, then it is cloudy. From the fact that if Elisabeth cooks, then it is cloudy, and that if it is cloudy, then Elisabeth is cooking we can infer that Elisabeth is cooking if and only if it is cloudy. From the fact that Elisabeth is cooking if and only if it is cloudy, and that it is not cloudy we can infer that Elisabeth is not cooking. From the fact that if Elisabeth cooks, then Barbara is playing a game, and that if Barbara plays a game, then Elisabeth is cooking we can infer that Elisabeth is cooking if and only if Barbara is playing a game. Finally, from the fact that Elisabeth is cooking if and only if Barbara is playing a game, and that Elisabeth is not cooking we can infer that Barbara is not playing a game. Therefore, the answer is yes.