You COULD mix with only headphones. But, you'llprobably find that even the best headphones won't give you a mix that will translate well anywhere else but on those phones.
As far as why you need headphones if you have speakers. ...You should do your mixing with speakers, but headphones are good for listening while doing surgical edits and more detailed work. But they're not absolutely necessary for mixing, you could definitely make a great mix with just speakers in a well treated room.