I just bought a 6d, i love it, I already had a 24-105 so I bought body only, I wanted to see how it handled low light, this photo was taken in a dark studio..almost no light. I just put in a+ mode, the green mode where the camera does everything.
this photo was also taken wirelessly with the eos android ap using my nexus4. At least i think it was..I was trying the ap out, 90% this pic was.. either way, the EOS ap is free, and works very well..
the camera chose the focus point, the closest point of course, the elbow.
iso 12,800 the exif is still attached
no adjustments just resize in lightroom and conversion from raw to jpg.
I'll let you decide,
I am not sure, but I think it is hard to tell fine detail on a down-sized-for-the-web photo. This was shot with a 60D, in about 5 stops less light than your 6D was using.
12,800 0.4s, f/5.6, 60D, 15-85. I had to go into the basement with the lights shut off, so I apologize for the content. It looks pretty good to me here. At 100% it is obviously noisy in the shadows. Useable? I think so.