In 1 Corinthians 11:3 Paul casually makes a really big statement about submission in marriage.
1 Corinthians 11:3 But I want you to understand that the head of every man is Christ, the head of a wife is her husband, and the head of Christ is God.
Notice that here he doesn’t simply say that the husband is the head of the household; he takes a giant step back and shows the bigger picture. So the husband is the head of the wife, but then Christ is the head of the man. And not only that, but God is the head of Christ.
Paul gives us a great analogy here for how the marriage relationship works: he compares it to the relation of God the Father to God the Son. Christ is, of course, equal in power and glory to his Father– he is the “radiance of the glory of God and the exact imprint of his nature” (Heb 1:3). But he also submits to his Father: for example, he does nothing on his own, but only what he sees the Father doing (John 5:19).
In other words, what we see in the Trinity (and it extends to the Holy Spirit as well) is total equality with regard to the persons— no person of the Trinity is more important or more divine than the others– with distinction in roles— the Son submits to the Father, and the Spirit to the Father and the Son. This fits very well with Paul’s teaching on marriage. Men and women are entirely equal in status before God, but there is a distinction in their individual roles when we come to the topic of marriage. So the same Paul who says “there is neither male nor female, for you are all one in Christ Jesus” (Gal 3:28) also has no problem saying “Wives, submit to your husbands” (Eph 5:22).
This isn’t a matter of men having wholesale authority over women; it’s a matter of every believer being properly submitted to God-given authority. Wives submit to husbands, husbands submit to Christ, and Christ submits to God.
Wiser Time 
the phrase that caught my attention in my Basic Christian Doctrine class (currently taking) was “mutual submission.” Whether it be people or the persons of the Trinity, both/all are willing to submit to the other’s authority and leading.
Paul does say that Christians should submit to each other, like in Eph 5:21. But in the same passage he shows what that looks like: servants submit to masters, wives to husbands, etc. Husbands are told to love their wives, care for their wives, understand their wives, but never to submit to their wives. Nor are fathers told to submit to their children, or masters to their servants.
I think the same thing’s true of the Trinity– I can’t (off the top of my head) think of anywhere where the Father is said to submit (or something similar) to the Son, or the Son to the Spirit. You’ll hear about the Father loving and honoring the Son, but not submitting to him. So I think egalitarians take the idea of “mutual submission” further than the Bible takes it.