Carpentry and Woodworking - Overhead Cabinet must go, please help

I just moved into a new house and there are way to many cabinets in the kitchen for me to even fill. There is a cabinet that needs to be taken out so my fridge will fit in its correct spot. I can not figure out how to take it down. The problem is to the right of the cabinet is a wall and to the other side is a set of larger cabinets. When looking at the cabinet that needs to come out there are 3 screws on each side at the doors drilling into the side wall and into the frame of the cabinets to the left. There are 4 more screws to the back going into the back wall. My problem is there are several nails going through the framing into both the wall and cabinets surrounding it. I have no way of prying these out from what i can tell. I don;t want to completely hack up the cabinets if I don't have to but I don;t have any other ideas. There is no leverage with space to try and even cut the nails with a small saw or anything. Any ideas how to get this out without damaging the wall or cabinets to its sides?

If you had a Dremel tool you could possibly cut the heads off the nails and then punch them through with a nail set. Also sometimes just using a hacksaw blade you can get into some pretty tight spaces. Good luck!

Another way might be to remove the cabinet to the left of it at the same time, separate them & rehang it.