Consider the following premises. For all x, x will write a letter if and only if x is a writer. For all x, x will write a letter or x is a poet. There is at least one x for which x is a writer or x is not a poet. Can we infer the following from them? For all x, x is not a writer.
 From the fact that there is at least one x for which x is a writer or x is not a poet we can infer that there is at least one x for which if x were a poet, then x is a writer. From the fact that for all x, x will write a letter if and only if x is a writer we can infer that for all x, if x writes a letter, then x is a writer. Finally, from the fact that for all x, if x writes a letter, then x is a writer, that there is at least one x for which if x were a poet, then x is a writer, and that for all x, x will write a letter or x is a poet we can infer that there is at least one x for which x is a writer, which contradicts that For all x, x is not a writer. Therefore, the answer is no.