Consider the following premises. If it is cloudy, then Thomas is cooking. Barbara is not playing a game. If it is windy, then it is cloudy. Barbara is playing a game if and only if it is windy. If Thomas cooks, then it is windy. Can we infer the following from them? If we can, name the inference rule being used: It is not cloudy.
 From the fact that if it is cloudy, then Thomas is cooking, and that if Thomas cooks, then it is windy we can infer that if it is cloudy, then it is windy via transitivity. From the fact that Barbara is playing a game if and only if it is windy, and that Barbara is not playing a game we can infer that it is not windy via biconditional elimination. From the fact that if it is cloudy, then it is windy, and that if it is windy, then it is cloudy we can infer that it is cloudy if and only if it is windy via biconditional introduction. Finally, from the fact that it is cloudy if and only if it is windy, and that it is not windy we can infer that it is not cloudy via biconditional elimination. Therefore, the answer is yes.