Best I can do is shoot off the hip and give you an honest guess. Maybe someone will come along that can do better.
It may work, but since the 10-sp chain is thinner, you might have a potential snag in your plans at the rear der locality as a 9-sp chain may not run nicely through there, if indeed at all. I'd consider it not likely. If it does, plan on the shifting being able to be adjusted to a reasonably ok quality but I would suspect it not to be 100% smooth.
For point in comparison, on my old bike from years ago that I refurbished and gave to a friend recently, it runs 8-sp shifters with a 9-sp chain and a 7-sp set of derailleurs. Chain width through those systems was the same and that part of the deal was ok, but even so the rear der could not be adjusted to run smooth and purring in all gears. In this case its because it only has 7-sp adjustment capabilities trying to shift through 8 locations. It works pretty well, but not perfect. Your setup might shift better due to the der being able to handle 10 shift locations and needing to cover 9, but again, only if the chain actually makes it through. My point being simply I wouldnt expect perfection if it works.
Try calling a local LBS for their thoughts, or if you have the parts, have them look at it or look and see yourself if the chain runs smoothly through the rear der.
Good luck!
