Consider the following premises. For all x_6, if x_6 climbs a mountain, then x_6 is a mechanic. For all x_6, if x_6 plays a game, then x_6 is climbing a mountain. For all x_6, if x_6 were a mechanic, then x_6 is playing a game. There is at least one x_6 for which if x_6 plays a game, then x_6 is climbing a mountain. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_6 for which x_6 will climb a mountain if and only if x_6 will play a game.
 From the fact that for all x_6, if x_6 climbs a mountain, then x_6 is a mechanic, and that for all x_6, if x_6 were a mechanic, then x_6 is playing a game we can infer that for all x_6, if x_6 climbs a mountain, then x_6 is playing a game via universal transitivity. From the fact that for all x_6, if x_6 climbs a mountain, then x_6 is playing a game, and that for all x_6, if x_6 plays a game, then x_6 is climbing a mountain we can infer that for all x_6, x_6 will climb a mountain if and only if x_6 will play a game via universal biconditional introduction. From the fact that for all x_6, x_6 will climb a mountain if and only if x_6 will play a game we can infer that for all x_6, if x_6 climbs a mountain, then x_6 is playing a game via universal biconditional elimination. Finally, from the fact that for all x_6, if x_6 climbs a mountain, then x_6 is playing a game, and that there is at least one x_6 for which if x_6 plays a game, then x_6 is climbing a mountain we can infer that there is at least one x_6 for which x_6 will climb a mountain if and only if x_6 will play a game via existential biconditional introduction. Therefore, the answer is yes.