Team Sky continued to control the pace in the bunch, with nobody else trying a move until a brief attack from Dan Martin (QuickStep Floors) just before the summit.
Featuring five climbs before a downhill finish, the stage was defined by the yellow jersey Greg van Avermaet (BMC) again having a good day in the saddle defending his position with a determined ride over his unfavoured terrain which earned him the jury award as the most aggressive rider of the day.
Speaking during The Bradley Wiggins Show by Eurosport, Sir Brad said that Brailsford would "certainly get in both their ears and make them both think that they can win it as a way of motivating them" and would be keen to keep both riders high on General Classification for as long as possible into the race until a natural selection occurs leaving one rider as outright team leader.
Sprinters keep suffering in Alps as Gaviria, Groenewegen abandon
However, Froome, who has won the last three Grand Tours, is happy to see how the race pans out before worrying about who will be the team's leader in the latter stages.
Thomas' Team Sky team-mate Chris Froome was third on the stage, level on time with Team Sunweb's Dumoulin as they came home 20 seconds after Thomas.
Julian Alaphilippe became the first Frenchman to win a stage in this year's Tour, claiming the 10th stage that included three first-category climbs and a beyond-category climb but ended with a descent and the contenders together in the peloton.More news: Microsoft Edge starts testing visual search with iOS beta users
Froome recovered quickly to rejoin teammate Thomas but the Team Sky riders had fallen nearly seven minutes behind Van Avermaet and the breakaway group.
Inside the final kilometre, Thomas kicked again and swept past Nieve with a couple of hundred metres remaining to the line as the Spaniard faded.
"To get a victory at the Tour de France was a dream for me", Alaphillipe said.More news: Does Putin have 'compromising material' on Trump? He denies it but
"I think it was a bit of a spur of the moment thing for us but it made sense, it was flawless, we didn't even have to talk", Froome said. It was a shame because Mikel Nieve is a nice guy but I had to go for the win. For me it's an unknown.
"To wear the yellow jersey is always a massive honour, " Thomas said. I managed to do it last year but to do it two years on the trot is really nice'.More news: Trump trolled with 'traitor' tweet by Dictionary.com after Putin summit
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