Consider the following premises. For all y_2, y_2 is a server or y_2 is not a carpenter. For all y_2, y_2 is a carpenter if and only if y_2 will go running. There is at least one y_2 for which if y_2 were a server, then y_2 is running. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y_2 for which y_2 is a server if and only if y_2 is a carpenter.
 From the fact that for all y_2, y_2 is a server or y_2 is not a carpenter we can infer that for all y_2, if y_2 were a carpenter, then y_2 is a server via universal rewriting. From the fact that for all y_2, y_2 is a carpenter if and only if y_2 will go running we can infer that for all y_2, if y_2 goes running, then y_2 is a carpenter via universal biconditional elimination. From the fact that there is at least one y_2 for which if y_2 were a server, then y_2 is running, and that for all y_2, if y_2 goes running, then y_2 is a carpenter we can infer that there is at least one y_2 for which if y_2 were a server, then y_2 is a carpenter via existential transitivity. Finally, from the fact that there is at least one y_2 for which if y_2 were a server, then y_2 is a carpenter, and that for all y_2, if y_2 were a carpenter, then y_2 is a server we can infer that there is at least one y_2 for which y_2 is a server if and only if y_2 is a carpenter via existential biconditional introduction. Therefore, the answer is yes.