Translate the following inference to logic notation: For all x, if x_8 sleeps, then x_8 is playing tennis. There is at least one x for which x_8 will sleep. For all x, x_8 will write a letter if and only if x_8 will play tennis. Therefore there is at least one x for which x_8 will write a letter.
forall x: P(x) -> Q(x). exists x: P(x). forall x: R(x) <-> Q(x). Therefore exists x: R(x).