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First make sure the clip in the Timeline is unlinked from the audio (Command l (lowercase L) toggles between linked and unlinked. Once the audio and video are unlinked, delete the video.

-DH
 
On the default interface, there is a button on the upper right corner of the timeline window, you can also use that to unlink the tracks.
 
At the left of the timeline, there are Padlock icons on each of the video/audio timelines. Select the clip you wish to delete(which also selects the audio), and lock the audio tracks you wish you keep, after it is locked, just delete the clip.

Just another option. :)
 
At the left of the timeline, there are Padlock icons on each of the video/audio timelines. Select the clip you wish to delete(which also selects the audio), and lock the audio tracks you wish you keep, after it is locked, just delete the clip.

Just another option. :)

I bet you started editing on Avid.
 
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