Consider the following premises. For all x_6, if x_6 were a janitor, then x_6 is an astronaut. For all x_6, if x_6 were a musician, then x_6 is an astronaut. For all x_6, x_6 will not cook or x_6 is a janitor. For all x_6, x_6 will cook or x_6 is a musician. For all x_6, x_6 will work. 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 astronaut or x_6 will work are not true.
 From the fact that for all x_6, x_6 will cook or x_6 is a musician, and that for all x_6, x_6 will not cook or x_6 is a janitor we can infer that for all x_6, x_6 is a musician or x_6 is a janitor via universal resolution. From the fact that for all x_6, if x_6 were a musician, then x_6 is an astronaut, that for all x_6, if x_6 were a janitor, then x_6 is an astronaut, and that for all x_6, x_6 is a musician or x_6 is a janitor we can infer that for all x_6, x_6 is an astronaut via universal disjunction elimination. Finally, from the fact that for all x_6, x_6 is an astronaut, and that for all x_6, x_6 will work we can infer that for all x_6, x_6 is an astronaut and x_6 will work via universal conjunction, which contradicts that There is at least one x_6 for which either x_6 is an astronaut or x_6 will work are not true. Therefore, the answer is no.