This is true because ideally it makes each person equally submissive to the other regardless of position. This makes equality a matter of disposition rather than a matter of position. The Bible does not teach that husbands are to have power over wives, nor wives power over husbands. It teaches mutual submission and love. This is true with every biblical concept of authority.
Submission requires sensitivity, openness, self-examination, flexibility and interest in others.
It requires treating others, as we want to be treated. If we would treat those under our authority as we treat those we are in authority too, we would see a big change in relationships. The biblical concept of love is to seek the highest good of the other person. When we can give up our positions, our possessions, and ourselves to seek the good of others, we become powerful servants in the army of God. I have always noticed that for those who want to serve there is always a position to be filled.