Moving a support post
There is no specific rule that tells you how deep you need to be. Typically the footing is sized based on the specific type of soil, i.e. organic soil is junk, and cannot be used, hard pack glacial till might be rated up to 6000 psf. You need to read the code, and check with your local code enforcement official. I assume you are pulling a permit, as this is a permanent structural modification to the house. Some codes set minimum depth for footings, some set minimum sizes.
As to the required depth, you don't kinow for a fact that the soil is undisturbed, your house could be built on fill. Once you open up the slab and are able to look at the soil, an experienced, qualified person (typically an engineer) can determine what the allowable bearing capacity of the soil is. Based on the load you are going to pick up, the qualified individual can then determine how big a footing, and how deep, it needs to be.