Consider the following premises. For all x_3, if x_3 were a lawyer, then x_3 is working. There is at least one x_3 for which if x_3 were a nurse, then x_3 is a lawyer. For all x_3, if x_3 works, then x_3 is a nurse. For all x_3, x_3 will not work. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_3 for which x_3 is not a nurse.
 From the fact that there is at least one x_3 for which if x_3 were a nurse, then x_3 is a lawyer, and that for all x_3, if x_3 were a lawyer, then x_3 is working we can infer that there is at least one x_3 for which if x_3 were a nurse, then x_3 is working via existential transitivity. From the fact that for all x_3, if x_3 works, then x_3 is a nurse, and that there is at least one x_3 for which if x_3 were a nurse, then x_3 is working we can infer that there is at least one x_3 for which x_3 will work if and only if x_3 is a nurse via existential biconditional introduction. Finally, from the fact that there is at least one x_3 for which x_3 will work if and only if x_3 is a nurse, and that for all x_3, x_3 will not work we can infer that there is at least one x_3 for which x_3 is not a nurse via existential biconditional elimination. Therefore, the answer is yes.