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That's an undistributed middle, which is a real logical fallacy. Your syllogism is invalid. Most people don't know logic, and very rarely are such fallacies invoked.
The point you missed is that "logical fallacy" never disproves something.
Shitty internet poster response is to claim that slippery slope "disproves" something but all it actually show is that it's not 100% proven. Yeah slippery slope means it doesn't go there 100% of the time, but when it's 80% instead that's hardly disproving it.