Translate the following inference to logic notation: For all x, if x_7 were a cashier, then x_7 is running. There is at least one x for which x_7 will not play a game. For all x, if x_7 goes running, then x_7 is a cashier. For all x, x_7 will play a game if and only if x_7 will go running. Therefore there is at least one x for which x_7 is not a cashier.
forall x: P(x) -> Q(x). exists x: ~R(x). forall x: Q(x) -> P(x). forall x: R(x) <-> Q(x). Therefore exists x: ~P(x).