Translate the following inference to logic notation: For all x, x_4 is a nurse. For all x, if x_4 writes a letter, then x_4 is not a nurse. There is at least one x for which x_4 is an astronaut if and only if x_4 will write a letter. Therefore there is at least one x for which x_4 is not an astronaut.
forall x: P(x). forall x: Q(x) -> ~P(x). exists x: R(x) <-> Q(x). Therefore exists x: ~R(x).