Consider the following premises. There is at least one x_2 for which if x_2 were sad, then x_2 is a bartender. For all x_2, x_2 is sad. For all x_2, x_2 is a poet. Can we infer the following from them? If we can, name the inference rule being used: For all x_2, either x_2 is a poet or x_2 is a bartender are not true.
 From the fact that there is at least one x_2 for which if x_2 were sad, then x_2 is a bartender, and that for all x_2, x_2 is sad we can infer that there is at least one x_2 for which x_2 is a bartender via existential modus ponens. Finally, from the fact that for all x_2, x_2 is a poet, and that there is at least one x_2 for which x_2 is a bartender we can infer that some poets are bartenders via existential conjunction, which contradicts that For all x_2, either x_2 is a poet or x_2 is a bartender are not true. Therefore, the answer is no.