Does mountain bike cycling translate into road/triathlon cycling fitness/strength?

Look at who won the Tour de France this past year - Cadel was a two time world cup champ on the mountain bike. I think that paid off quite well for him.

As long as you're working hard on the bike, it doesn't matter if it's a road bike, mountain bike, or beach cruiser, it will help!

I have never tried a mountain bike, but when you are riding, how different are the various angles (knees and hips)  relative to a road bike ?

Mikki Weiss is another one that seemed to do well at the mountain bike to tri switch.

That was my first argument to my buddy & training partner.  Weiss, Llanos, Dibens, ...  My buddy and training partner said it was not conclusive evidence.

When I am seated on my mountain bike I "feel" myself using some different muscle groups (using my calves and lower legs more) and my hip angle is much different than when I'm on my tri Bike.  However, when I climb I like getting out of the saddle as my HR stays low and I can generate power easier and I "feel" like I am using the same exact muscles wether climbing on a mountain bike or tri bike.  But again this is all anectodal as I am not certain. 

I am going to continue to ride hilly/challenging mountain bike courses as I enjoy riding them and believe that this will translate into improved climbing ability on my tri bike; i'd just like some concrete evidence that proves it.  As for mountain biking on technical, rooty trails; I am not a fan. 

I'm curious why they say there isn't. You're still pedaling in a circle. The closer the body angles are the more you're using the muscles in the same way between the bikes. We say that all the time for road to tri bikes and there is no reason this won't apply to mountain bikes as well.

You did mention standing up to pedal the hills. Standing positions might vary considerably, so there could be some truth, but how much depends on the variance. And if you're going to stand on the mountain bike vs sit on the tri-bike, then yeah, you might notice a difference as there would be a considerable difference in position.

Now if the trails are very technical it will be more difficult to push yourself as hard. I can't take myself as far into "that place" on the mountain bike simply because I need mental energy to steer much more precisely than when on the roads. You may also be limited on how hard you can continually push yourself due to technical sections being skill limited vs power limited. That's why the pro mountain bikers will ride the road bikes a couple times a week. It's easier for them to build fitness. And again, another pro example of switching between bike types.

It's no different from translating road bike riding to triathlon bike riding.  Sure you may have some slight changes like hip angle, but you're still riding a bicycle.  The only real difference would be how you train, you don't get as many simple 2x20 FTP intervals on a mountain bike.