Well the nature of an emergency means that one needs the money straight away. There's no guarantee that someone will loan the money at a future date. Also there is the security involved in knowing that a rainy day fund exists. Personally, I'd never leave myself without money in the bank. Then again I don't have any loans except mortgage and would save for anything I needed. That wasn't always the case when I was younger though.
If you don't mind me asking roughly how much is your emergency fund. We keep about €2k in the bank for this purpose as we both have PHI, income continuance protection, and life assurance.
One of us recently inherited about €50k which we used to reduce the mortgage but some people were advising us to hold onto some of it.