Translate the following inference to logic notation: There is at least one x for which x_9 is sad if and only if x_9 is an astronaut. For all x, x_9 will cook if and only if x_9 is an astronaut. Therefore there is at least one x for which if x_9 were sad, then x_9 is cooking.
exists x: P(x) <-> Q(x). forall x: R(x) <-> Q(x). Therefore exists x: P(x) -> R(x).