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In the first set Federer broke Robredo at the first opportunity before closing out the opener with little fuss.
Although Robredo won more games in the second, he didn't win a single point on Federer's serve as the seven-time champion cruised through.
The third set was much tighter, with both men made to work on serve before a costly forehand error at the net from Robredo led to Federer clinching the all-important break.
Robredo had a break-back point when 5-4 down in the third, but couldn't take it and soon Federer converted his third match point to seal victory.
Federer could face fellow Swiss and the man seeded directly behind him at number five, Stanislas Wawrinka, in the last eight.