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I assume you're talking about the actual physical prophecy in the Department of Mysteries, those are pretty much like 'video' documentation of the prophecy so nothing would happen if they were smashed beyond losing like a backup copy of what Dumbledore remembers from the event. The reason it was so important to Voldemort is because he only heard part of the prophecy and was desperate to hear the rest so that he could try to prevent his death.

Dumbledore's explanation to Harry was that Voldemort wanted to hear the entire prophecy so that he could learn how to kill Harry.

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