Consider the following premises. For all x_6, if x_6 plays a game, then x_6 is a carpenter. For all x_6, if x_6 were fast, then x_6 is playing a game. There is at least one x_6 for which x_6 is a carpenter if and only if x_6 is fast. Can we infer the following from them? There is at least one x_6 for which x_6 is fast if and only if x_6 is a carpenter.
 From the fact that for all x_6, if x_6 were fast, then x_6 is playing a game, and that for all x_6, if x_6 plays a game, then x_6 is a carpenter we can infer that for all x_6, if x_6 were fast, then x_6 is a carpenter. From the fact that there is at least one x_6 for which x_6 is a carpenter if and only if x_6 is fast we can infer that there is at least one x_6 for which if x_6 were a carpenter, then x_6 is fast. Finally, from the fact that for all x_6, if x_6 were fast, then x_6 is a carpenter, and that there is at least one x_6 for which if x_6 were a carpenter, then x_6 is fast we can infer that there is at least one x_6 for which x_6 is fast if and only if x_6 is a carpenter. Therefore, the answer is yes.