Consider the following premises. For all x_9, if x_9 were a carpenter, then x_9 is excited. For all x_9, if x_9 were excited, then x_9 is a carpenter. For all x_9, x_9 is excited or x_9 is not a carpenter. Can we infer the following from them? If we can, name the inference rule being used: For all x_9, x_9 is excited if and only if x_9 is a carpenter.
We can infer this via the universal biconditional introduction rule. Therefore, the answer is yes.