Translate the following inference to logic notation: For all x, if x_8 were a nurse, then x_8 is a carpenter. There is at least one x for which if x_8 were a carpenter, then x_8 is a politician. For all x, x_8 is a politician if and only if x_8 is a nurse. Therefore there is at least one x for which x_8 is a nurse if and only if x_8 is a carpenter.
forall x: P(x) -> Q(x). exists x: Q(x) -> R(x). forall x: R(x) <-> P(x). Therefore exists x: P(x) <-> Q(x).