Consider the following premises. There is at least one x_6 for which x_6 will sleep or x_6 will make tea. For all x_6, x_6 will not sleep or x_6 will go running. For all x_6, if x_6 makes tea, then x_6 is working. For all x_6, x_6 will go running if and only if x_6 will work. Can we infer the following from them? If we can, name the inference rule being used: For all x_6, x_6 will not work.
 From the fact that there is at least one x_6 for which x_6 will sleep or x_6 will make tea, and that for all x_6, x_6 will not sleep or x_6 will go running we can infer that there is at least one x_6 for which x_6 will make tea or x_6 will go running via existential resolution. From the fact that for all x_6, x_6 will go running if and only if x_6 will work we can infer that for all x_6, if x_6 goes running, then x_6 is working via universal biconditional elimination. Finally, from the fact that for all x_6, if x_6 makes tea, then x_6 is working, that for all x_6, if x_6 goes running, then x_6 is working, and that there is at least one x_6 for which x_6 will make tea or x_6 will go running we can infer that there is at least one x_6 for which x_6 will work via existential disjunction elimination, which contradicts that For all x_6, x_6 will not work. Therefore, the answer is no.