I changed my Apple ID/email address without logging out on my iphone

I understand now that I should have logged out of all my Apple ID locations before changing my email address, however now that I have changed the old email address in iTunes and not on my phone, I have a problem.

The iphone keeps asking me to log in on the old email address/ID but because it no longer exists, I can't do that, and it won't let me log out without logging in.

Does anyone know how I rectify this problem and install the new details onto the iPhone/

Many thanks.