Djokovic overcame a series of rows with the chair umpire to record a 6-3 3-6 6-2 6-2 victory over Japan's former US Open finalist.
Federer was bounced out in the quarterfinals in five sets. The top-seed breezed into the quarterfinals and won by 6-0 7-5 6-4. We thought he could lose to maybe somebody else, but not necessarily Anderson, who hasn't had huge success coming into this tournament. I just kept on telling myself, 'I have to keep believing.' I kept saying that today was going to be my day, because you really need that mindset taking the court against somebody like Roger.
Two unforced errors from Anderson handed Federer the first break within the first two minutes and the Swiss opened up a 3-1 lead with a quick-fire ace.
The first hiccup surfaced at the start of the second set with Anderson breaking the Federer serve and ending that run of holds.
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"I think it was great quality tennis and in the final set there were some unbelievable points", said 2008 and 2010 champion Nadal. After that I never really felt exactly 100 %. Not only has he yet to drop a set through four matches at the All-England Club this year, but he hasn't had his serve broken, and it wasn't until his last match against Adrian Mannarino that he faced his first break point.
Eighth seed Anderson will play 2016 runner-up Milos Raonic or American ninth seed John Isner on Friday for a place in Sunday's final. Despite starting the game at 4-5 brightly (40-15), Anderson found himself in a bit of trouble when a forehand error saw him go down a match point against Federer.
Wednesday's other quarter-final sees 2016 runner-up Raonic tackle USA ninth seed Isner, making his Wimbledon last-eight debut at 33. At 33 years old, he'd taken just one of the previous 18 Slams and the chorus of questions about whether he'd ever win another seemed to be reaching its crescendo.
Isner rallied behind a dominant first serve, converting 73 percent of them and winning those serves at an 80 percent clip.
Raonic has the fastest serve of this year's Wimbledon at 147 miles (236.5 kilometres) per hour, while Isner is just behind on 144mph (231.7kph).
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Djokovic will play Nadal in the semifinals.
He said: "Beating Federer at Wimbledon will be one I'm going to remember, especially in such a close match". Djokovic last appeared in the semifinals at Wimbledon in 2015, when he won. After making it to the fourth round of the Australian Open, an event where he's dominated historically, Djokovic lost in the first round of several masters events, including the Indian Wells Masters and the Miami Open.
Djokovic has played twice on the 11,000-capacity Court One this year but was shunted off to the 4,000-seater Court Two for his second round clash against Horacio Zeballos. "I feel like I'm peaking at the right moment". It was a bad look for a player who's usually gracious (or at least gives the appearance of it) in defeat.
But he found himself in even more unfamiliar territory with an inspired Anderson's relentless, thudding serve eventually overpowering the 20-time grand slam victor.
Anderson never had made it beyond the fourth round at Wimbledon until this week, carried by his booming serve.
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