Translate the following inference to logic notation: For all x, if x were an astronaut, then x is a soldier. For all x, if x were an astronaut, then x is working. There is at least one x for which if x were a soldier, then x is working. For all x, if x works, then x is an astronaut. Therefore there is at least one x for which x is an astronaut if and only if x will work.
forall x: P(x) -> Q(x). forall x: P(x) -> R(x). exists x: Q(x) -> R(x). forall x: R(x) -> P(x). Therefore exists x: P(x) <-> R(x).