Consider the following premises. For all x, if x were happy, then x is a carpenter. For all x, if x were happy, then x is a carpenter. For all x, if x were a carpenter, then x is happy. Can we infer the following from them? If we can, name the inference rule being used: For all x, x is a carpenter if and only if x is happy.
 From the fact that for all x, if x were happy, then x is a carpenter, and that for all x, if x were a carpenter, then x is happy we can infer that for all x, x is happy if and only if x is a carpenter via universal biconditional introduction. From the fact that for all x, x is happy if and only if x is a carpenter we can infer that for all x, if x were a carpenter, then x is happy via universal biconditional elimination. Finally, from the fact that for all x, if x were a carpenter, then x is happy, and that for all x, if x were happy, then x is a carpenter we can infer that for all x, x is a carpenter if and only if x is happy via universal biconditional introduction. Therefore, the answer is yes.