A motorbike braked hard around 1.2 kilometres to race with the fans crowding in and Richie Porte (BMC Racing) collided into it. Froome, in yellow, and Bauke Mollema (Trek-Segafredo) fell too.
Froome was left without a bike and at one point ran up the mountain to try to keep pace with the race leaders.
Yates was behind Froome at the time of the crash, but passed Froome and Porte in its aftermath. He finished 11th behind Mollema in 10th – enough to take the overall race leader in the provisional standings.
"I don't want to take the jersey like that. I'd rather take it with my legs and not a crash in a bad situation," Yates said. "Everyone saw it. I don't think anyone would want to take it that way. If I was in the same situation in the yellow jersey, I'd want the same outcome.
The provisional classification marked Yates in the lead by nine seconds over Mollema, 14 seconds over Nairo Quintana (Movistar) and 28 seconds over Romain Bardet (AG2R La Mondiale). Froome sat sixth at 53". Around 30 minutes later, a new classification appeared with Froome on top of Yates by 47 seconds and Mollema by 56 seconds.
"Froome is stronger than me and the rightful owner of the yellow jersey,” Yates added. “It wouldn't felt right to take the jersey like that. If anyone else was in the same situation, they'd feel the same."
Fans filled the climb, meant to be one of the Tour's crown stages on Bastille Day. High winds had already forced the organiser ASO to shorten it by six kilometres, but this was still expected to be a race-defining stage for the general classification. Shortly before the line, the festive day turned into a journée noire for ASO.
"It was pretty dangerous [with the fans], but the fans make the sport special," 23-year-old Yates added. "There are not many sports where the fans can get so close."
Yates still maintains the white jersey of best young rider by 1-42 minutes over South African Louis Meintjes (Lampre-Merida).
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