Translate the following inference to logic notation: For all x, x_2 is a bartender or x_2 is not an astronaut. There is at least one x for which if x_2 were a bartender, then x_2 is an astronaut. Therefore there is at least one x for which x_2 is an astronaut if and only if x_2 is a bartender.
forall x: P(x) or ~Q(x). exists x: P(x) -> Q(x). Therefore exists x: Q(x) <-> P(x).