Consider the following premises. For all x_15, if x_15 were a carpenter, then x_15 is a cashier. For all x_15, x_15 is a carpenter if and only if x_15 is sad. For all x_15, if x_15 plays a game, then x_15 is sad. There is at least one x_15 for which if x_15 were a cashier, then x_15 is playing 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 a carpenter.
 From the fact that for all x_15, x_15 is a carpenter if and only if x_15 is sad we can infer that for all x_15, if x_15 were sad, then x_15 is a carpenter via universal biconditional elimination. From the fact that for all x_15, if x_15 were a carpenter, then x_15 is a cashier, and that there is at least one x_15 for which if x_15 were a cashier, then x_15 is playing a game we can infer that there is at least one x_15 for which if x_15 were a carpenter, then x_15 is playing a game via existential transitivity. From the fact that for all x_15, if x_15 plays a game, then x_15 is sad, and that for all x_15, if x_15 were sad, then x_15 is a carpenter we can infer that for all x_15, if x_15 plays a game, then x_15 is a carpenter via universal transitivity. From the fact that there is at least one x_15 for which if x_15 were a carpenter, then x_15 is playing a game, and that for all x_15, if x_15 plays a game, then x_15 is a carpenter we can infer that there is at least one x_15 for which x_15 is a carpenter 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 a carpenter 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 a carpenter via existential biconditional elimination. Therefore, the answer is yes.