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The Scotsman struggled to get a stronghold of the game in the early phases of the first set after Federer nabbed an early break to take control of the court.
While Murray showed some impressive movement rushing to the net at times, the former world number one was on top of his game once again as he displayed well-placed and swift shots across the court.
After taking a one-set lead, Federer was able to score 16 winners and a break to dominate once again, while Murray couldn't find a way through.
Despite scoring four aces and looking strong at times during his service games, the Wimbledon champion was unable to create a break-point opportunity of his own and relinquished the second set.
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With Federer needing just one more set to wrap up the match, Murray had it all to do, but once again games went by the way of serve until the Swiss pro converted one of five break points.
It left Federer serving for the match, but Murray managed to break back at a crucial time and draw the third set level on five games all.
Despite Murray coming to life, Federer was able to hold serve to love to take the set into a tie-break, but Murray bounced back once again to save two match points to take the third set.
After 10 deuces in the second gruelling game of the fourth set, Murray was able to maintain his serve and prevent Federer from taking a break at the first time of asking.
From then on, both players managed to hold serve, but the Scotsman was pushed to the limit during most of his games, while Federer appeared to get through his with relative ease.
Murray began to show signs of struggle following his impressive comeback in the third, and Federer was able to pounce to grab a vital break and serve once again for the match.
Despite some fightback from Murray, Federer was able to close the match and set up a semi-final against Rafael Nadal.