Consider the following premises. For all x, if x were fast, then x is playing a game. For all x, if x plays a game, then x is fast. There is at least one x for which x is fast if and only if x will play a game. For all x, x will not play a game. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x for which x is not fast.
 From the fact that for all x, if x plays a game, then x is fast, and that for all x, if x were fast, then x is playing a game we can infer that for all x, x will play a game if and only if x is fast via universal biconditional introduction. From the fact that there is at least one x for which x is fast if and only if x will play a game we can infer that there is at least one x for which if x plays a game, then x is fast via existential biconditional elimination. From the fact that for all x, x will play a game if and only if x is fast we can infer that for all x, if x were fast, then x is playing a game via universal biconditional elimination. From the fact that for all x, if x were fast, then x is playing a game, and that there is at least one x for which if x plays a game, then x is fast we can infer that there is at least one x for which x is fast if and only if x will play a game via existential biconditional introduction. Finally, from the fact that there is at least one x for which x is fast if and only if x will play a game, and that for all x, x will not play a game we can infer that there is at least one x for which x is not fast via existential biconditional elimination. Therefore, the answer is yes.