Consider the following premises. For all x_13, x_13 will write a letter. For all x_13, x_13 will sleep or x_13 will not take a plane. There is at least one x_13 for which x_13 will take a plane. Can we infer the following from them? If we can, name the inference rule being used: For all x_13, x_13 will not sleep.
 From the fact that for all x_13, x_13 will sleep or x_13 will not take a plane we can infer that for all x_13, if x_13 takes a plane, then x_13 is sleeping via universal rewriting. From the fact that for all x_13, if x_13 takes a plane, then x_13 is sleeping, and that there is at least one x_13 for which x_13 will take a plane we can infer that there is at least one x_13 for which x_13 will sleep via existential modus ponens. From the fact that there is at least one x_13 for which x_13 will sleep, and that for all x_13, x_13 will write a letter we can infer that there is at least one x_13 for which x_13 will sleep and x_13 will write a letter via existential conjunction. Finally, from the fact that there is at least one x_13 for which x_13 will sleep and x_13 will write a letter we can infer that there is at least one x_13 for which x_13 will sleep via existential simplification, which contradicts that For all x_13, x_13 will not sleep. Therefore, the answer is no.