"I'm happy to be in the second week, quarter-finals and now the semi-finals".
Kerber, who spent 34 weeks at world number one in 2016 and 2017, could be set for a rematch with Williams if the seven-time champion can get past Julia Goerges in the second semi-final.
She faces Latvian 12th seed Jelena Ostapenko, a former French Open champion, for a place in the semi-finals.
Playing her first Wimbledon semifinal, 21-year-old Ostapenko rallied to save a match point and break Kerber's serve for the first time, but the German saved a break point as she served out the match at the second attempt.
The 52nd-ranked Giorgi of Italy took the first set 6-3.
That tunnel vision allowed the 30-year-old to become the only woman to reach at least the quarter-finals in all three of this year's majors.
Williams won Wimbledon on her last two visits in 2015 and 2016 before missing last year's event due to her pregnancy. "I'm going to go out there and just do what I can, do my best, see what happens". "I think everyone here has a great chance so you have to take it", Goerges said.
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The government has reportedly struggled to offer exact information about the number of children separated from their parents. For almost one month Perez knew nothing about her children - "the longest, most torturous time of my life", she said.
It is the first time in the Open era, which began in 1968, that two German women have reached the Wimbledon semi-finals in the same year. "She didn't let me play how I wanted", said Cibulkova, the world number 33.
The young Latvian has been her fiery self so far this Wimbledon, and has been particularly ruthless on return, winning 20 of 38 return games on her opponents' serve. Just going to prepare well for the match. "I have all this motivation to just win another one".
Kasatkina was unable to force a third set, though, drilling a forehand into the base of the net at full stretch to end her best run at Wimbledon.
The seven-time champion, who missed the tournament previous year, was outplayed by Italian Camila Giorgi in the first set but rallied to win 3-6 6-3 6-4 on Centre Court.
Ostapenko came out on the offensive, converting her sixth break point to take an early lead, but she was immediately pegged back by the Slovak who broke to love before securing the first hold of the match. "I think I finished kind of confident".
Two more consecutive breaks of serve came in games four and five as the crowd, for the first time in the fortnight wrapped up against the chilly conditions, revelled in a true 50:50 match-up.
Asked if she had any sympathy for Hsieh, as replays suggested the Taiwan player deserved to have the point replayed, Cibulkova said: "No".
Men's quarterfinals: Kevin Anderson beat Roger Federer 2-6, 6-7 (5), 7-5, 6-4, 13-11; John Isner beat Milos Raonic 6-7 (5), 7-6 (7), 6-4, 6-3; Novak Djokovic beat Kei Nishikori 6-3, 3-6, 6-2, 6-2; Rafael Nadal beat Juan Martin del Potro 7-5, 6-7 (7), 4-6, 6-4, 6-4.