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Not going nowhere does not necessarily imply that you "are going" somewhere. You could be somewhere that is not nowhere already.
You could be not going nowhere because you are already nowhere and you stay there
I am not going nowhere is always true.
You never go nowhere, you don’t go, or you go somewhere.
You could just not be going.
You would then say you either:
1. Aren't going anywhere.
2. Are going nowhere.
A double negative in the English language, which this question is fundamentally about, evaluates to a positive.
Your proof is a bit poor.
Logically your statement means: "It is not the case that I'm going nowhere".
"I'm going nowhere" can be paraphrased as "I'm not going anywhere".
If it is put under negation we get "It is not the case that I'm not going anywhere".
Now, this implies going somewhere.
We may not be talking about standard English. If we were to take AAVE (African American Vernacular English) into account for example, it would, indeed mean "I'm not going anywhere"