Ask your bank / lender.
Some won't bridge unless the mortgage is already discharged (which is a pain), others will bridge but may require that some of the bridging is used to discharge any remaining mortgage. Most will provide open ended bridging only on a low loan to value (<50%), so if you're relying on the full value of your current house to complete the transaction it can be a problem. If you have the flexibility it's a great way to go - means you don't have the dual headache of house-selling and timing househunting and purchase to coincide, you're in a no-chain situation, and your offers won't be ignored because you don't have contracts signed. But if you need to realise all your equity for the new purchase up front, or are living in a slower-to-sell area, I'm afraid there's probably not much option but to bite the bullet, put your property on the market, and hope you can choreograph the whole transaction... Good luck.