Nadal's knees have struggled with the move from clay to grass in recent years but he has breezed through the draw so far and secured another straight-sets win, beating Czech Jiri Vesely 6-3 6-3 6-4.
The world number one said: "Of course it's an important result for me. It's true that it has been a while since I have been in that position but at the same time I have been in that position a couple of times in my career. We are not talking about already that I played a final or I won the tournament. When I come here, I come here thinking I can do a good result."
Federer has been equally emphatic and, after taking just 16 minutes to win the opening set against Adrian Mannarino, he at least found himself in a match before coming through 6-0 7-5 6-4.
Federer extended his run of consecutive sets won at Wimbledon to 32 and still has not had his serve broken all tournament, although he was forced to save his first break points.
The first set was so one-sided that, when Mannarino trailed 0-40 in the opening game of the second, the normally pro-Federer Centre Court crowd began to cheer loudly for his opponent.
It had the potential to be embarrassing, particularly given Mannarino is a top-30 player, but the Frenchman composed himself well to make a match of it.
Federer said: "I was also surprised it was that fast, that first set, especially 16 minutes. That was too fast. Shouldn't really happen, but thankfully they do for me. I probably won't have another 6-0 set this week, so I'll enjoy this one."
In the last eight on Wednesday, Federer will face eighth seed Kevin Anderson, one of a number of players enjoying a career-best Wimbledon after a 7-6 (7/4) 7-6 (7/2) 5-7 7-6 (7/4) victory over Gael Monfils.
John Isner and Kei Nishikori are also through to the last eight for the first time. Isner has only reached one previous grand slam quarter-final, at the US Open in 2011, and his serve proved too strong for 19-year-old Stefanos Tsitsipas in a 6-4 7-6 (12/10) 7-6 (7/4) victory.
Isner is hoping he might have some presidential support should he reach the semi-finals, with Donald Trump due to visit the UK later this week.
"I'd love to have Trump come watch me," said Isner, a supporter of the Republican Party. "That would be awesome. Maybe I'll tweet at him if I win on Wednesday. I know a lot of people won't like that, but I don't care."
Next up for the American will be a battle of the big servers against Milos Raonic, who overcame Isner's countryman Mackenzie McDonald 6-3 6-4 6-7 (5/7) 6-2 to make the last eight for the fourth time in five years.
Nishikori missed both the US Open and Australian Open with wrist problems but now appears close to his best again and, thanks to a 4-6 7-6 (7/5) 7-6 (12/10) 6-1 victory over Ernests Gulbis, becomes the first Japanese man to make the last eight at all four slams.
