Consider the following premises. For all x_5, if x_5 were a writer, then x_5 is making tea. For all x_5, x_5 will listen to a song. For all x_5, x_5 is excited or x_5 is a writer. For all x_5, if x_5 were excited, then x_5 is making tea. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_5 for which x_5 will not make tea.
 From the fact that for all x_5, if x_5 were excited, then x_5 is making tea, that for all x_5, if x_5 were a writer, then x_5 is making tea, and that for all x_5, x_5 is excited or x_5 is a writer we can infer that for all x_5, x_5 will make tea via universal disjunction elimination. From the fact that for all x_5, x_5 will make tea, and that for all x_5, x_5 will listen to a song we can infer that for all x_5, x_5 will make tea and x_5 will listen to a song via universal conjunction. Finally, from the fact that for all x_5, x_5 will make tea and x_5 will listen to a song we can infer that for all x_5, x_5 will make tea via universal simplification, which contradicts that There is at least one x_5 for which x_5 will not make tea. Therefore, the answer is no.