Consider the following premises. David is not fast. If David were fast, then David is happy. For all x, if x were happy, then x is fast. If Barbara were a janitor, then David is happy. If Barbara plays a game, then Barbara is a janitor. Can we infer the following from them? If we can, name the inference rule being used: Barbara is not playing a game.
 From the fact that for all x, if x were happy, then x is fast we can infer that if David were happy, then David is fast via universal instantiation. From the fact that if David were fast, then David is happy, and that if David were happy, then David is fast we can infer that David is fast if and only if David is happy via biconditional introduction. From the fact that if Barbara plays a game, then Barbara is a janitor, and that if Barbara were a janitor, then David is happy we can infer that if Barbara plays a game, then David is happy via transitivity. From the fact that David is fast if and only if David is happy, and that David is not fast we can infer that David is not happy via biconditional elimination. Finally, from the fact that if Barbara plays a game, then David is happy, and that David is not happy we can infer that Barbara is not playing a game via modus tollens. Therefore, the answer is yes.