Translate the following inference to logic notation: For all x, if x_5 were a doctor, then x_5 is sleeping. For all x, if x_5 sleeps, then x_5 is a doctor. There is at least one x for which if x_5 were a doctor, then x_5 is sleeping. Therefore there is at least one x for which x_5 is a doctor if and only if x_5 will sleep.
forall x: P(x) -> Q(x). forall x: Q(x) -> P(x). exists x: P(x) -> Q(x). Therefore exists x: P(x) <-> Q(x).