Translate the following inference to logic notation: For all x, if x_6 were a poet, then x_6 is happy. For all x, if x_6 were happy, then x_6 is writing a letter. There is at least one x for which x_6 is a poet or x_6 will not write a letter. Therefore there is at least one x for which x_6 is a poet if and only if x_6 will write a letter.
forall x: P(x) -> Q(x). forall x: Q(x) -> R(x). exists x: P(x) or ~R(x). Therefore exists x: P(x) <-> R(x).