Fight it. You have a good case and the phone records can prove it. More to the point the officers have to prove that you WERE using the phone. If they can't, they're in big trouble. They also have to prove that you have more than one phone. 

Curiously, as of today the police are coming down hard in the UK on drivers using their phones while driving. It's £60 sterling and 3 points on your license if you're caught. (12 points and you lose your license) 
I personnly take issue with drivers using their phones while driving. it's bloody dangerous and we're getting to the stage over here where it is becoming socially unacceptable to do this and I expect that we'll have a 'GET OFF THE PHONE' campagne(SS) pretty soon. I always feel like yelling this at guilty drivers now.

However, getting back to your incident, I can see this happening more and more as the cops realise that they can meet their arrest targets by hitting on drivers with false accusations. In the UK the car-owner is considered an easy target by the governement. We are taxed to the hilt by them in the name of road-safety, congestion and pollution. Our speed cameras are nothing but money-making machines under the name of 'SPEED KILLS' when in fact speed is the factor in only about 10% of all road deaths.

Take this one to court and set a preceedent for all those wrongfully accused drivers to come.
Good luck!

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All the Woes of a World by Jonathan Icknield aka The Bryster

And in my final hours - I would cling rather to the tattooed hand of kindness - than the unblemished hand of hate...