Translate the following inference to logic notation: For all x, if x_7 climbs a mountain, then x_7 is excited. For all x, x_7 is a cashier or x_7 will sleep. There is at least one x for which x_7 is not a cashier or x_7 will climb a mountain. For all x, x_7 will sleep if and only if x_7 is excited. Therefore there is at least one x for which x_7 is excited.
forall x: P(x) -> Q(x). forall x: R(x) or S(x). exists x: ~R(x) or P(x). forall x: S(x) <-> Q(x). Therefore exists x: Q(x).