Consider the following premises. There is at least one y_4 for which if y_4 were a janitor, then y_4 is a lawyer. For all y_4, if y_4 were a lawyer, then y_4 is working. For all y_4, y_4 is a janitor or y_4 will not work. 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 will work if and only if y_4 is a lawyer.
 From the fact that for all y_4, y_4 is a janitor or y_4 will not work we can infer that for all y_4, if y_4 works, then y_4 is a janitor via universal rewriting. From the fact that for all y_4, if y_4 works, then y_4 is a janitor, and that there is at least one y_4 for which if y_4 were a janitor, then y_4 is a lawyer we can infer that there is at least one y_4 for which if y_4 works, then y_4 is a lawyer via existential transitivity. Finally, from the fact that there is at least one y_4 for which if y_4 works, then y_4 is a lawyer, and that for all y_4, if y_4 were a lawyer, then y_4 is working we can infer that there is at least one y_4 for which y_4 will work if and only if y_4 is a lawyer via existential biconditional introduction. Therefore, the answer is yes.