Consider the following premises. For all x_6, x_6 will cook or x_6 will drive a car. For all x_6, if x_6 takes a plane, then x_6 is an office clerk. For all x_6, x_6 will take a plane. For all x_6, x_6 will not cook. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_6 for which either x_6 is an office clerk or x_6 will drive a car are not true.
 From the fact that for all x_6, x_6 will cook or x_6 will drive a car, and that for all x_6, x_6 will not cook we can infer that for all x_6, x_6 will drive a car via universal disjunctive syllogism. From the fact that for all x_6, if x_6 takes a plane, then x_6 is an office clerk, and that for all x_6, x_6 will take a plane we can infer that for all x_6, x_6 is an office clerk via universal modus ponens. Finally, from the fact that for all x_6, x_6 is an office clerk, and that for all x_6, x_6 will drive a car we can infer that for all x_6, x_6 is an office clerk and x_6 will drive a car via universal conjunction, which contradicts that There is at least one x_6 for which either x_6 is an office clerk or x_6 will drive a car are not true. Therefore, the answer is no.