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