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