Consider the following premises. For all x_10, x_10 is a cashier if and only if x_10 is a carpenter. For all x_10, x_10 is not a carpenter. There is at least one x_10 for which if x_10 were fast, then x_10 is sad. For all x_10, if x_10 were sad, then x_10 is fast. For all x_10, x_10 is a cashier if and only if x_10 is sad. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_10 for which x_10 is not fast.
 From the fact that for all x_10, x_10 is a cashier if and only if x_10 is a carpenter, and that for all x_10, x_10 is not a carpenter we can infer that for all x_10, x_10 is not a cashier via universal biconditional elimination. From the fact that there is at least one x_10 for which if x_10 were fast, then x_10 is sad, and that for all x_10, if x_10 were sad, then x_10 is fast we can infer that there is at least one x_10 for which x_10 is fast if and only if x_10 is sad via existential biconditional introduction. From the fact that for all x_10, x_10 is a cashier if and only if x_10 is sad, and that for all x_10, x_10 is not a cashier we can infer that for all x_10, x_10 is not sad via universal biconditional elimination. Finally, from the fact that there is at least one x_10 for which x_10 is fast if and only if x_10 is sad, and that for all x_10, x_10 is not sad we can infer that there is at least one x_10 for which x_10 is not fast via existential biconditional elimination. Therefore, the answer is yes.