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