That's what I use with my 7D. My wife uses the 7D + 400/5.6L.
AF speed and sharpness wise, the 400/5.6L is better, sometimes significantly. You just need to learn to deal with it.
I can keep up with the warblers and flight-shots just fine. I also get shots that my wife can't due to the much shorter minimum focal distance (1.5m vs the 3.5m of the 400/5.6L). I can also shoot in darker conditions with IS on and using flash, I find it helps quite a bit.
If the subject is less than a 1/4 frame, then the 400/5.6L wins easily in detail captured which will cover most things over 20ft away. But if you did not have both to compare, you won't really notice.
I also have a Sigma 50-500, Sigma 120-300/2.8, Sigma 400/5.6 APO, and the EF 300/4L IS is easily a match for all of them.
A big difference, is to use the Canon 1.4xTC mark II version. I've used the Canon mark 1 version, the Sigma 1.4, a Tamron 1.4 (4 element only), and AF speed & sharpness is easily better with the Mark II 1.4x TC.