To say "I rest my case" means therefore that, as far as you are concerned, you've done more than enough to prove your point, and need say no more.
John: "You know your problem, Bill? You're far too quick to descend to insulting people who disagree with you."
: Bill: "Don't be ridiculous! That's the sort of thing only a complete idiot would say."
: John: "I rest my case..."
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