Consider the following premises. For all x_7, x_7 is a doctor if and only if x_7 is a carpenter. There is at least one x_7 for which if x_7 were a lawyer, then x_7 is a carpenter. For all x_7, x_7 will drive a car if and only if x_7 is a doctor. For all x_7, x_7 will not drive a car. 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 is not a lawyer.
 From the fact that for all x_7, x_7 will drive a car if and only if x_7 is a doctor, and that for all x_7, x_7 will not drive a car we can infer that for all x_7, x_7 is not a doctor via universal biconditional elimination. From the fact that for all x_7, x_7 is a doctor if and only if x_7 is a carpenter, and that for all x_7, x_7 is not a doctor we can infer that for all x_7, x_7 is not a carpenter via universal biconditional elimination. Finally, from the fact that there is at least one x_7 for which if x_7 were a lawyer, then x_7 is a carpenter, and that for all x_7, x_7 is not a carpenter we can infer that there is at least one x_7 for which x_7 is not a lawyer via existential modus tollens. Therefore, the answer is yes.