Translate the following inference to logic notation: For all x, z is an astronaut if and only if z is a bartender. For all x, if z were an astronaut, then z is a bartender. Therefore for all x, if z were a bartender, then z is an astronaut.
forall x: P(x) <-> Q(x). forall x: P(x) -> Q(x). Therefore forall x: Q(x) -> P(x).