Vidic has been at Old Trafford for eight years, making 296 appearances to date, but will leave at the end of the current season and join Inter.
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"Nemanja has been here a long time and produced so many great performances and won a lot of trophies," Smalling told United's official club magazine.
"He's set the benchmark in terms of being a real defensive force and, although it won't be easy, we'll do our best to fill that void.
"One thing he has always told me is 'defend first'.
"When you're defending, it's always nice to clear a ball by playing it out with a nice pass or set us away on an attack, but at the end of the day you're a defender and your job, first and foremost, is to defend.
"Therefore, you have to stop whoever you're playing against, no matter what."