Consider the following premises. For all y_4, if y_4 were poor, then y_4 is a carpenter. There is at least one y_4 for which if y_4 were poor, then y_4 is a carpenter. For all y_4, if y_4 were a carpenter, then y_4 is poor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y_4 for which y_4 is poor if and only if y_4 is a carpenter.
 From the fact that for all y_4, if y_4 were a carpenter, then y_4 is poor, and that there is at least one y_4 for which if y_4 were poor, then y_4 is a carpenter we can infer that there is at least one y_4 for which y_4 is a carpenter if and only if y_4 is poor via existential biconditional introduction. From the fact that there is at least one y_4 for which y_4 is a carpenter if and only if y_4 is poor we can infer that there is at least one y_4 for which if y_4 were a carpenter, then y_4 is poor via existential biconditional elimination. Finally, from the fact that for all y_4, if y_4 were poor, then y_4 is a carpenter, and that there is at least one y_4 for which if y_4 were a carpenter, then y_4 is poor we can infer that there is at least one y_4 for which y_4 is poor if and only if y_4 is a carpenter via existential biconditional introduction. Therefore, the answer is yes.