Translate the following inference to logic notation: For all x, if x_7 works, then x_7 is writing a letter. There is at least one x for which x_7 is a nurse or x_7 will not write a letter. For all x, if x_7 were a nurse, then x_7 is working. Therefore there is at least one x for which x_7 will write a letter if and only if x_7 is a nurse.
forall x: P(x) -> Q(x). exists x: R(x) or ~Q(x). forall x: R(x) -> P(x). Therefore exists x: Q(x) <-> R(x).