"That game, 5-4 in the second set, affected me a little bit", Nadal said in his on-court interview.
"I'm going to play against the victor of this match [between Djokovic and Zverev] and both of them are great", said Thiem.
Tsitsipas' record against Nadal now stands at 1-6, with his only victory coming, somewhat unexpectedly, on clay in Madrid last spring.
World number one Novak Djokovic and Alexander Zverev are to face off tomorrow to decide who plays Thiem.
In the second match, Rafa suffered a tight 7-6, 7-6 loss to Dominic Thiem, squandering chances in both sets and setting a must-winning clash with Stefanos Tsitsipas in the battle for a place in the semis.
Hitting a service victor in nearly 40% of all the points, Rafa had a massive 44-28 advantage in the shortest rallies up to four strokes, standing in the court on the second-serve return and earning those two breaks that carried him over the top. Playing only 14 matches since March due to coronavirus pandemic, Nadal was fresh and rested ahead of the premium ATP event, the one that has kept running away from him ever since the debut in 2006. "I think past year he was very exhausted mentally at the end of the season". "Dominic didn't start that well and I took advantage [from] the beginning".
Thiem said the loss to Rublev was "not painful at all".
76ers trading Al Horford, picks to Thunder for Danny Green
The Sixers get major cap flexibility by taking Green with just $15 million and one more season left on his deal. Adams' numbers were down this past season, averaging 10.9 points and 9.3 rebounds in the regular season.
The already-qualified Thiem was short of energy, too, when play eventually began. I was able to find a way to win.
"I think I was quite comfortable until that moment, after that everything changed a bit, I started to serve a little bit worse but at the end a very positive match for me and for me to be in the semi-finals is another important thing".
"With the fact that I was already qualified in the back of my mind, it was maybe hard today to keep that intensity alive", said Thiem, who had already clinched top spot in Group London 2020.
Rublev grabbed an early break in the second set but faltered at 4-3 when a netted forehand allowed Thiem to level.
But Rublev broke the Austrian world number three again at 5-5 with a yet another forehand victor and finished the job a game later to record a third straight win over Thiem. "I think he (was) focused on the semis". Dominating with his initial shot, Nadal had 16 winners and 11 unforced errors, leaving the Russian on 19 winners and 23 mistakes. "He won that game and then I was starting to suffer with my serve".
"It was not easy for Dommy", said Rublev, who has won five events on the tour this year in a breakthrough season. That helped me to have more confidence and helped me to play better and better...