Consider the following premises. For all x_6, x_6 will sleep or x_6 will write a letter. For all x_6, if x_6 writes a letter, then x_6 is driving a car. For all x_6, x_6 will drive a car or x_6 will not sleep. For all x_6, x_6 will play squash. 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 will play squash or x_6 will drive a car are not true.
 From the fact that for all x_6, x_6 will drive a car or x_6 will not sleep we can infer that for all x_6, if x_6 sleeps, then x_6 is driving a car via universal rewriting. From the fact that for all x_6, if x_6 sleeps, then x_6 is driving a car, that for all x_6, if x_6 writes a letter, then x_6 is driving a car, and that for all x_6, x_6 will sleep or x_6 will write a letter we can infer that for all x_6, x_6 will drive a car via universal disjunction elimination. Finally, from the fact that for all x_6, x_6 will play squash, and that for all x_6, x_6 will drive a car we can infer that for all x_6, x_6 will play squash 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 will play squash or x_6 will drive a car are not true. Therefore, the answer is no.