Consider the following premises. For all x_5, if x_5 listens to a song, then x_5 is cooking. For all x_5, if x_5 goes running, then x_5 is a poet. For all x_5, if x_5 cooks, then x_5 is running. For all x_5, x_5 will listen to a song. 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 is not a poet.
 From the fact that for all x_5, if x_5 cooks, then x_5 is running, and that for all x_5, if x_5 goes running, then x_5 is a poet we can infer that for all x_5, if x_5 cooks, then x_5 is a poet via universal transitivity. From the fact that for all x_5, if x_5 listens to a song, then x_5 is cooking, and that for all x_5, x_5 will listen to a song we can infer that for all x_5, x_5 will cook via universal modus ponens. Finally, from the fact that for all x_5, if x_5 cooks, then x_5 is a poet, and that for all x_5, x_5 will cook we can infer that for all x_5, x_5 is a poet via universal modus ponens, which contradicts that There is at least one x_5 for which x_5 is not a poet. Therefore, the answer is no.