Consider the following premises. For all x_15, if x_15 plays a game, then x_15 is fast. For all x_15, if x_15 were fast, then x_15 is climbing a mountain. There is at least one x_15 for which if x_15 climbs a mountain, then x_15 is playing tennis. For all x_15, x_15 will play tennis if and only if x_15 will 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_15 for which if x_15 plays a game, then x_15 is fast.
 From the fact that for all x_15, x_15 will play tennis if and only if x_15 will play a game we can infer that for all x_15, if x_15 plays tennis, then x_15 is playing a game via universal biconditional elimination. From the fact that there is at least one x_15 for which if x_15 climbs a mountain, then x_15 is playing tennis, and that for all x_15, if x_15 plays tennis, then x_15 is playing a game we can infer that there is at least one x_15 for which if x_15 climbs a mountain, then x_15 is playing a game via existential transitivity. From the fact that for all x_15, if x_15 were fast, then x_15 is climbing a mountain, and that there is at least one x_15 for which if x_15 climbs a mountain, then x_15 is playing a game we can infer that there is at least one x_15 for which if x_15 were fast, then x_15 is playing a game via existential transitivity. From the fact that there is at least one x_15 for which if x_15 were fast, then x_15 is playing a game, and that for all x_15, if x_15 plays a game, then x_15 is fast we can infer that there is at least one x_15 for which x_15 is fast if and only if x_15 will play a game via existential biconditional introduction. Finally, from the fact that there is at least one x_15 for which x_15 is fast if and only if x_15 will play a game we can infer that there is at least one x_15 for which if x_15 plays a game, then x_15 is fast via existential biconditional elimination. Therefore, the answer is yes.