Translate the following inference to logic notation: For all x, if x_11 were a politician, then x_11 is cooking. For all x, if x_11 were a bartender, then x_11 is a politician. There is at least one x for which x_11 will cook if and only if x_11 is a bartender. For all x, if x_11 cooks, then x_11 is a bartender. Therefore there is at least one x for which x_11 is a bartender if and only if x_11 will cook.
forall x: P(x) -> Q(x). forall x: R(x) -> P(x). exists x: Q(x) <-> R(x). forall x: Q(x) -> R(x). Therefore exists x: R(x) <-> Q(x).