This is too hypothetical to comment upon. Questions:
How much do you have in the emergency fund? If you have only $3000, you shouldn't deplete it by half -- it'd take you too low. On the other hand, if you have $30,000, removing half would still give you a good cushion.
How fast could you replenish the emergency fund? Are you the person who will actually do that, or would you plan to do it and never get around to it?
Do you have any reason to suspect that you'd need that emergency fund in the near future? Job insecure? Going to need to replace your furnace or roof? Of course, we never know when we're going to have an emergency, but sometimes we know that expensive things are coming up in the future. IF you see any of these things on the horizon, it might be a time to hold onto cash. (For example, I know that this time next year I"m going to start paying for braces -- I know I'm going to dip into my short-term savings.)
And though it's been asked before, how are you paying $80/month in interest on a car? Do you have something super-expensive, or is your rate uber-high? Can you afford this car? I'm suspicious that I couldn't.
I got a better deal on a car (just bought a 2011 Nissan Juke on Saturday) when I told them I was paying cash for it. Its one less bill I have to worry about paying each month.
I bought my car three years ago, but my experience was just the opposite; They were rather disappointed when I paid cash. It meant that they weren't going to make interest money from me month after month after month.
For the record, when we bought my car three years ago, we faced just this question! We could pay cash by essentially depleting our short-term savings and checking account . . . or we could finance a portion of the car. We didn't have to touch our emergency savings, but we bought the car weeks before a relatively expensive vacation and two months before Christmas . . . but my old car was DEAD, and we had to make the purchase THEN. We did choose to empty out our short-term savings and checking. We had a couple lean months, but we went into it knowing that we'd made that choice. We bought the car in late October, and by February we were "ourselves" again. A couple paychecks, eating out of the pantry instead of grocery shopping, and our checking account looked good again. Given the same circumstances, I'd make the same choice again. We both agreed that avoiding financing was worth those couple lean months, and we were both very motivated to build our balances up again.