Translate the following inference to logic notation: For all x, if z writes a letter, then z is a bartender. There is at least one x for which if z were a bartender, then z is writing a letter. Therefore there is at least one x for which z is a bartender if and only if z will write a letter.
forall x: P(x) -> Q(x). exists x: Q(x) -> P(x). Therefore exists x: Q(x) <-> P(x).