Translate the following inference to logic notation: There is at least one x for which z_4 is a soldier if and only if z_4 is sad. For all x, if z_4 were a soldier, then z_4 is cooking. For all x, if z_4 cooks, then z_4 is sad. Therefore there is at least one x for which z_4 is a soldier if and only if z_4 is sad.
exists x: P(x) <-> Q(x). forall x: P(x) -> R(x). forall x: R(x) -> Q(x). Therefore exists x: P(x) <-> Q(x).