^Exactly. This is known to some as the "zero-carb" diet:
Zero-Carb Diet - Wikipedia, the free encyclopedia
I'm not advocating this idea at all. Just pointing out that it is possible and you won't have problems with nutritional deficiencies. If you do the opposite and try to cut out fats or proteins completely, you're in for some bad news. Now, I'm sure this type of diet could increase the risk for many dieases in the long run (but it can also reduce the symptoms of others like diabetes).