"It's very kind of you to look after the baby all day."
"Oh, that's all right, it was nothing, I enjoy it."
In this case, could nothing be replaced by not a thing? What's the difference between nothing and not a thing in general?
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I think that "not a thing" is used for specifically referring to an actual set of items, whereas in your example it is not.
In these sentences, I have replaced "nothing" with "not a thing".
Correct: Not a thing was stolen when I left the shop unlocked.
Correct: Not a thing is preventing someone from getting in.
Incorrect: I've installed a new lock for you. Don't thank me; it was not a thing.
I feel that "nothing" in the context of politely refusing praise cannot be replaced by "not a thing".
"not a thing" can be expanded to "not a single a thing" and "not even a single thing" for emphasis.