An LTI system is said to be minimum-phase if the system and its inverse system are causal and stable. That's implying that all poles and zeros must be strictly inside the unit circle.
An all-pass filter's zero-pole pairs are reflections of each other across the unit circle so that an all-pass filter absorbs the poles or zeros. For a stable all-pass filter, its zeros are outside the unit circle and its poles are inside the unit circles.
A causal and stable mixed-phase system has all poles inside the unit circle and zeros which could be inside or outside the unit circle.
Therefore, reflecting zeros outside the unit circle by cascading an all-pass system makes all zeros inside the unit circle, which means you get a minimum-phase system. (As Dan pointed out in the comment, cascading an all pass with a mixed-phase system is incorrect. Extracting an all pass from the mixed phase results in a minimum phase, or in the contrary cascading an all pass with a minimum phase results in a mixed-phase system.)
Once you get a minimum-phase system, the only thing you can do to prevent changing its magnitude response is to cascade an all-pass system. That brings additional zeros outside the unit circle and the system is no longer minimum-phase. Therefore minimum-phase system with a specific magnitude response is unique.