Consider the following premises. For all x_12, x_12 is a janitor. For all x_12, x_12 will cook or x_12 will listen to a song. For all x_12, if x_12 listens to a song, then x_12 is driving a car. There is at least one x_12 for which x_12 will not cook or x_12 is rich. For all x_12, if x_12 were rich, then x_12 is driving a car. Can we infer the following from them? If we can, name the inference rule being used: For all x_12, either x_12 is a janitor or x_12 will drive a car are not true.
 From the fact that for all x_12, x_12 will cook or x_12 will listen to a song, and that there is at least one x_12 for which x_12 will not cook or x_12 is rich we can infer that there is at least one x_12 for which x_12 will listen to a song or x_12 is rich via existential resolution. From the fact that for all x_12, if x_12 listens to a song, then x_12 is driving a car, that for all x_12, if x_12 were rich, then x_12 is driving a car, and that there is at least one x_12 for which x_12 will listen to a song or x_12 is rich we can infer that there is at least one x_12 for which x_12 will drive a car via existential disjunction elimination. Finally, from the fact that for all x_12, x_12 is a janitor, and that there is at least one x_12 for which x_12 will drive a car we can infer that some janitors will drive a car via existential conjunction, which contradicts that For all x_12, either x_12 is a janitor or x_12 will drive a car are not true. Therefore, the answer is no.