Consider the following premises. For all x_4, if x_4 were a poet, then x_4 is a politician. For all x_4, if x_4 were happy, then x_4 is a poet. There is at least one x_4 for which x_4 is a writer or x_4 is an office clerk. For all x_4, x_4 is a poet or x_4 is not a writer. For all x_4, if x_4 were an office clerk, then x_4 is happy. Can we infer the following from them? If we can, name the inference rule being used: For all x_4, x_4 is not a politician.
 From the fact that for all x_4, if x_4 were an office clerk, then x_4 is happy, and that for all x_4, if x_4 were happy, then x_4 is a poet we can infer that for all x_4, if x_4 were an office clerk, then x_4 is a poet via universal transitivity. From the fact that for all x_4, x_4 is a poet or x_4 is not a writer we can infer that for all x_4, if x_4 were a writer, then x_4 is a poet via universal rewriting. From the fact that for all x_4, if x_4 were a writer, then x_4 is a poet, that for all x_4, if x_4 were an office clerk, then x_4 is a poet, and that there is at least one x_4 for which x_4 is a writer or x_4 is an office clerk we can infer that there is at least one x_4 for which x_4 is a poet via existential disjunction elimination. Finally, from the fact that for all x_4, if x_4 were a poet, then x_4 is a politician, and that there is at least one x_4 for which x_4 is a poet we can infer that there is at least one x_4 for which x_4 is a politician via existential modus ponens, which contradicts that For all x_4, x_4 is not a politician. Therefore, the answer is no.