Federer, who had played his last 20 matches on Centre Court, stretching back to 2015, continued to enjoy the full backing of his ever-growing fandom on Court No.1.
"I really hope it's an example of sticking to your dreams and keep believing in yourself". I said today is going to be my day. That's the mindset you need against someone like Roger.
"I feel like a lot of aspects of my game can give him a lot of trouble", Anderson said. "Matches likes that are very special".
Nadal reached the quarterfinals of Wimbledon for the first time since 2011 after defeating Jiri Vesely in a 6-3, 6-3, 7-5 victory on Monday.
Federer seemed to be heading for a routine victory with match point to complete a straights set win in the third.
Yet Anderson, the eighth seed, was meant to pose a far greater threat to Federer than his previous four opponents, and so it proved.
As the fifth set became as much a test of mental strength as anything, from 4-all to 6-all to 8-all to 10-all, Anderson was as steady as he needed to be.
Djokovic secured that game with a forehand victor, then saved two game points on Nishikori's serve before converting his first match point with a forehand down the line.
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Anderson stayed in the match thanks to his serving, compiling 28 aces.
This is Djokovic's second consecutive Wimbledon quarterfinal. He was, after all, the runner-up at last year's U.S. Open.
But his run at Wimbledon might have come as a surprise even to Anderson, who had never made it past the fourth round prior to this year and lost in the first round of the grass-court event at Queen's Club last month.
Normal service was resumed for a while at least as Federer held to love before immediately breaking back and taking the ensuing tie-break.
But Anderson was just getting started.
Federer was completely out of sorts by now and his uncharacteristic struggles on his forehand didn't improve in the fourth set. After that, Federer began to struggle and Anderson unloaded his swings on every shot.
That took great self-belief, he added, knowing that he had never taken a set off 20-time major victor Federer in four previous meetings. Anderson upset top seed Roger Federer of Switzerland on Wednesday.
Anderson, the first South African to reach the last eight since Wayne Ferreira in 1994, was steadfast though and did what no man had done at Wimbledon since last year's semi-final when he broke Federer's serve early in the second set.
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