The New Testament speaks about submission for all Christ's followers

Ephesians 5: 21 "Submit to one another, out of reverence for Christ"

This command goes without saying, that it is just not about women submitting to their husbands, but that husbands too are to submit to their wives. For all are equal in the eyes of Christ.

Phillipians 2:1-11) 1 Therefore if you have any encouragement from being united with Christ, if any comfort from his love, if any common sharing in the Spirit, if any tenderness and compassion, 2 then make my joy complete by being like-minded, having the same love, being one in spirit and of one mind. 3 Do nothing out of selfish ambition or vain conceit. Rather, in humility value others above yourselves, 4 not looking to your own interests but each of you to the interests of the others.
5 In your relationships with one another, have the same mindset as Christ Jesus:

6 Who, being in very nature[a] God,
did not consider equality with God something to be used to his own advantage;
7 rather, he made himself nothing
by taking the very nature of a servant,
being made in human likeness.
8 And being found in appearance as a man,
he humbled himself
by becoming obedient to death—
even death on a cross!

9 Therefore God exalted him to the highest place
and gave him the name that is above every name,
10 that at the name of Jesus every knee should bow,
in heaven and on earth and under the earth,
11 and every tongue acknowledge that Jesus Christ is Lord,
to the glory of God the Father.

We should follow Christ's example to be a servant to others, men and women and children alike. Make yourselves humble and submit to eachother and serve one another for in humility begins the path to righteousness.

John 13: 12-17) 12 When he had finished washing their feet, he put on his clothes and returned to his place. “Do you understand what I have done for you?” he asked them. 13 “You call me ‘Teacher’ and ‘Lord,’ and rightly so, for that is what I am. 14 Now that I, your Lord and Teacher, have washed your feet, you also should wash one another’s feet. 15 I have set you an example that you should do as I have done for you. 16 Very truly I tell you, no servant is greater than his master, nor is a messenger greater than the one who sent him. 17 Now that you know these things, you will be blessed if you do them.

Should women be treated equals with men? Jesus saw fit to have Mary his mother and the other unnamed women present in the upper room at Pentecost where they recieved the Holy Spirit and became Apostles of Christ along with the men. Both men and women alike were chosen to serve him equally for no one was less in his eyes than any other as all have sinned.

Acts 1: 13-14) 13And when they had entered, they went up to the upper room, where they were staying, Peter and John and James and Andrew, Philip and Thomas, Bartholomew and Matthew, James the son of Alphaeus and Simon the Zealot and Judas the son of James. 14All these with one accord were devoting themselves to prayer, together with the women and Mary the mother of Jesus, and his brothers.

If the women hadn't been chosen to preach equally as the men they would have never been present at Pentecost to recieve the Holy Spirit that empowered them to go out and preach the gospel boldly to the world.

I'm not sure what your point is, unless you're arguing that women and men should have equal rights in Christ.

If that's the case, then you have no issue with the Bible, which clearly teaches that men and women have equal rights under Christ. Now, if you're trying to argue that their ROLES should be the same, then you'll have an issue.

You can only argue for equality in mens and womens roles if you ignore large portions of Scripture. You, for example, have left out significant passages in the books to Timothy that talk about the role of women in the church.

So either you accept all of Scripture, and accept that men and women have different roles, or you reject elements of Scripture, in which case there will be no agreement between you and I... for I believe that "All Scripture is God-Breathed, and is useful for teaching, rebuking, correcting and training in righteousness".

If you yourself claim to agree with that verse, then you'll need to take ALL of scripture and find a logical reconciliation of those verses that claim that men and women are equal, AND those that state that women are not to teach men or preach over men in the church...

and not teach one at the exclusion of the other.

This is only possible if you accept that men and women are equal in rights, but different in their roles, and different in what God has called them to do.