Translate the following inference to logic notation: There is at least one x for which if x_1 were a doctor, then x_1 is running. For all x, if x_1 sleeps, then x_1 is a doctor. For all x, if x_1 goes running, then x_1 is sleeping. Therefore there is at least one x for which x_1 will go running if and only if x_1 is a doctor.
exists x: P(x) -> Q(x). forall x: R(x) -> P(x). forall x: Q(x) -> R(x). Therefore exists x: Q(x) <-> P(x).