  The first two of these are forms of AC for \(X\); while classically equivalent, in IST AC\(^*_X\) implies AC\(_X\), but not conversely. The principles DAC\(_X\) and DAC\(^*_X\) are dual forms of the axiom of choice for \(X\): classically they are both equivalent to AC\(_X\) and AC\(^*_X\) but intuitionistically DAC\(^*_X\) implies DAC\(_X\), and not conversely.