Consider the following premises. There is at least one x_7 for which if x_7 were a politician, then x_7 is a musician. For all x_7, x_7 will drive a car. For all x_7, x_7 is a politician or x_7 will not drive a car. Can we infer the following from them? If we can, name the inference rule being used: For all x_7, x_7 is not a musician.
 From the fact that for all x_7, x_7 is a politician or x_7 will not drive a car we can infer that for all x_7, if x_7 drives a car, then x_7 is a politician via universal rewriting. From the fact that for all x_7, if x_7 drives a car, then x_7 is a politician, and that there is at least one x_7 for which if x_7 were a politician, then x_7 is a musician we can infer that there is at least one x_7 for which if x_7 drives a car, then x_7 is a musician via existential transitivity. Finally, from the fact that there is at least one x_7 for which if x_7 drives a car, then x_7 is a musician, and that for all x_7, x_7 will drive a car we can infer that there is at least one x_7 for which x_7 is a musician via existential modus ponens, which contradicts that For all x_7, x_7 is not a musician. Therefore, the answer is no.