Consider the following premises. For all x_10, x_10 will not drive a car. For all x_10, if x_10 were a mechanic, then x_10 is a musician. For all x_10, if x_10 were a musician, then x_10 is a mechanic. For all x_10, x_10 is a mechanic if and only if x_10 will work. For all x_10, if x_10 drives a car, then x_10 is working. For all x_10, if x_10 works, then x_10 is driving a car. Can we infer the following from them? If we can, name the inference rule being used: For all x_10, x_10 is not a musician.
 From the fact that for all x_10, if x_10 works, then x_10 is driving a car, and that for all x_10, if x_10 drives a car, then x_10 is working we can infer that for all x_10, x_10 will work if and only if x_10 will drive a car via universal biconditional introduction. From the fact that for all x_10, if x_10 were a mechanic, then x_10 is a musician, and that for all x_10, if x_10 were a musician, then x_10 is a mechanic we can infer that for all x_10, x_10 is a mechanic if and only if x_10 is a musician via universal biconditional introduction. From the fact that for all x_10, x_10 will work if and only if x_10 will drive a car, and that for all x_10, x_10 will not drive a car we can infer that for all x_10, x_10 will not work via universal biconditional elimination. From the fact that for all x_10, x_10 is a mechanic if and only if x_10 will work, and that for all x_10, x_10 will not work we can infer that for all x_10, x_10 is not a mechanic via universal biconditional elimination. Finally, from the fact that for all x_10, x_10 is a mechanic if and only if x_10 is a musician, and that for all x_10, x_10 is not a mechanic we can infer that for all x_10, x_10 is not a musician via universal biconditional elimination. Therefore, the answer is yes.