Translate the following inference to logic notation: For all x, x_8 is a cashier or x_8 is not happy. There is at least one x for which x_8 is a cashier if and only if x_8 is happy. Therefore there is at least one x for which x_8 is happy if and only if x_8 is a cashier.
forall x: P(x) or ~Q(x). exists x: P(x) <-> Q(x). Therefore exists x: Q(x) <-> P(x).