I don't agree. If you took that to the extreme, and had an "only fat" diet, you would lose muscle because of a lack of protein, especially if you exercise.
Well, I was assuming that the OP wasn't taking it to the extreme ... but perhaps I should not assume such things.
You've got to have a balance ... approx. 20 to 35 percent of your caloric intake should be fat, approx. 45 to 65 percent of your caloric intake should be carbs, and whatever's left over should be protein.
But what I was answering was the underlying question about whether the fat we eat and the fat we burn are the same thing. And they are not the same thing ... at least not directly. It would appear that the OP is under the impression that the fat a person consumes goes right into fat cells on the body ... and therefore if we stop consuming fat, or drastically reduce the amount of fat we eat, we would lose "fat weight". This is a misconception of what happens when food enters our bodies