Consider the following premises. For all x_7, x_7 is a writer or x_7 is excited. For all x_7, x_7 is excited if and only if x_7 will climb a mountain. There is at least one x_7 for which if x_7 climbs a mountain, then x_7 is a poet. For all x_7, if x_7 were a writer, then x_7 is climbing a mountain. Can we infer the following from them? If we can, name the inference rule being used: For all x_7, x_7 is not a poet.
 From the fact that for all x_7, x_7 is excited if and only if x_7 will climb a mountain we can infer that for all x_7, if x_7 were excited, then x_7 is climbing a mountain via universal biconditional elimination. From the fact that for all x_7, if x_7 were a writer, then x_7 is climbing a mountain, that for all x_7, if x_7 were excited, then x_7 is climbing a mountain, and that for all x_7, x_7 is a writer or x_7 is excited we can infer that for all x_7, x_7 will climb a mountain via universal disjunction elimination. Finally, from the fact that there is at least one x_7 for which if x_7 climbs a mountain, then x_7 is a poet, and that for all x_7, x_7 will climb a mountain we can infer that there is at least one x_7 for which x_7 is a poet via existential modus ponens, which contradicts that For all x_7, x_7 is not a poet. Therefore, the answer is no.