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Williams, a six-time US Open winner, romped to a 6-3, 6-4 victory over Croatian 22nd seed Petra Martic to set up a quarter-final with China's Wang Qiang, the conqueror of world number two Barty.
Djokovic, champion in four of the past five Slams and 16 overall, was less bothered by left shoulder pain that nagged him in the second round in defeating 111th-ranked American Denis Kudla 6-3, 6-4, 6-2.
Djokovic was treated for a sore left shoulder throughout his 6-4, 7-6 (7/3), 6-1 victory over Argentina's 56th-ranked Juan Ignacio Londero.
Osaka was trailing American Sofia Kenin in the third set of their quarterfinal in Cincinnati when she sought treatment on her heavily-strapped left knee. After briefly returning to play, the Japanese star retired with Kenin leading 6-4, 1-6, 2-0.
Halep is through to the last four at Wimbledon for the first time since 2014 after a 7-6(4), 6-1 victory over China's Zhang Shuai.
Halep and Azarenka were among four former WTA number ones in singles action on day three.
Kerber began her title defence by surviving a battle of wits at Wimbledon as she beat fellow German Tatjana Maria on Centre Court.
Gauff, the youngest woman ever to survive the qualifying tournament, played like a seasoned performer as she clinically beat the 39-year-old great 6-4 6-4 to earn a standing ovation from an enchanted Court One crowd.
Japsn's Osaka was beaten by Czech Katerina Siniakova, the world’s top-ranked doubles player, 6-4, 6-2.
The powerful Czech struggled throughout to find the range on her feared serve and forehand as 31st seed Martic matched her best performance by reaching the last 16.