If you teach long enough, you will make public mistakes every now and then. Don't worry about it. You are human.
I have heard of teachers who were terrified of making a mistake in front of their classes. It really bothered them. It's just not worth worrying about. Put your pride aside and learn to deal with mistakes. They happen.
Stay calm when you make a mistake, correct yourself, maybe even laugh about it. Move on. It's no big deal. You don't have to get everything right to be an authority.