Consider the following premises. For all x_7, x_7 is not rich. For all x_7, if x_7 were rich, then x_7 is working. There is at least one x_7 for which if x_7 works, then x_7 is rich. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_7 for which x_7 will not work.
 From the fact that for all x_7, if x_7 were rich, then x_7 is working, and that there is at least one x_7 for which if x_7 works, then x_7 is rich we can infer that there is at least one x_7 for which x_7 is rich if and only if x_7 will work via existential biconditional introduction. From the fact that there is at least one x_7 for which x_7 is rich if and only if x_7 will work we can infer that there is at least one x_7 for which if x_7 works, then x_7 is rich via existential biconditional elimination. Finally, from the fact that there is at least one x_7 for which if x_7 works, then x_7 is rich, and that for all x_7, x_7 is not rich we can infer that there is at least one x_7 for which x_7 will not work via existential modus tollens. Therefore, the answer is yes.