Chemicals like sodium lauryl sulfate and other fatty acids with organosulfate head groups, which are much more powerful surfactants than the fatty acid sodium salts you get by reacting lye with a fat (like vegetable oil). "Traditional" soaps like that also contain glycerol (formed when the lye cleaves the glycerol backbone off of a triglyceride), which acts as a humectant moisturizer.
Technically, at least in the US, chemicals like SLS aren't legally classified as soap, and must be called a detergent. Which is why so many products are called things like "body wash" and "body bar", and you wont find the word "soap" on their packaging.
Same thing that would happen if you tried to lift your friend while they tried to lift you.
Or, to remove the layer of indirection: grab your own feet and lift up. Same thing as that.