Team Sky's Thomas came home 21 seconds behind Spain's Bilbao (Astana), who staged a heroic solo effort 12 kilometres from the finish to triumph.
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"I think these four days are the hardest four days I've raced in a one-week stage race", said Thomas.
"Without the boys here I would not have won the Dauphine". I can rest a little bit and wait for July.
Yates' victory was his second on the WorldTour this season, after also winning a stage at Tirreno-Adriatico, as he powered clear of Thomas and Romain Bardet (Ag2r-La Mondiale) in the final 200m. Bilbao made his move late on during the gruelling stage that is nearly the same as stage 11 in the Tour de France next month, with four major climbs and a summit finish at la Rosiere. I get to ride the races I want and I am a leader in a lot of them.
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England's Adam Yates, who's twin Simon wore the pink jersey at the Giro d'Italia for 13 stages in May, was third on the day at 15 seconds. "They're super hard climbing stages and even harder over the weekend".
It was Thomas' second World Tour stage race win after he won Paris-Nice in 2016.
Thomas managed to close the gap on the peloton within 15km, but was far from happy with the conduct of AG2R as he informed Romain Bardet when he caught up with the French rider.
"It's been a learning process starting in a new team so getting this win for UAE is great", he said.
Having been the last man standing in support of Thomas, meanwhile, Tao Geoghegan Hart climbed to 13th overall to follow his fifth place in support of Egan Arley Bernal's Amgen Tour of California win last month.