Consider the following premises. There is at least one x_6 for which if x_6 listens to a song, then x_6 is making tea. For all x_6, if x_6 sleeps, then x_6 is making tea. For all x_6, if x_6 writes a letter, then x_6 is sleeping. For all x_6, if x_6 goes running, then x_6 is listening to a song. For all x_6, x_6 will go running or x_6 will write a letter. Can we infer the following from them? If we can, name the inference rule being used: For all x_6, x_6 will not make tea.
 From the fact that for all x_6, if x_6 writes a letter, then x_6 is sleeping, and that for all x_6, if x_6 sleeps, then x_6 is making tea we can infer that for all x_6, if x_6 writes a letter, then x_6 is making tea via universal transitivity. From the fact that for all x_6, if x_6 goes running, then x_6 is listening to a song, and that there is at least one x_6 for which if x_6 listens to a song, then x_6 is making tea we can infer that there is at least one x_6 for which if x_6 goes running, then x_6 is making tea via existential transitivity. Finally, from the fact that there is at least one x_6 for which if x_6 goes running, then x_6 is making tea, that for all x_6, if x_6 writes a letter, then x_6 is making tea, and that for all x_6, x_6 will go running or x_6 will write a letter we can infer that there is at least one x_6 for which x_6 will make tea via existential disjunction elimination, which contradicts that For all x_6, x_6 will not make tea. Therefore, the answer is no.