Typically, 4 is by definition 1+(1+(1+1))), so "2+2=4" means

(1+1)+(1+1) = 1+(1+(1+1)))

In other words, it means two particular different processes for adding up 1's will yield the same result. This is not assumed in Peano arithmetic, but proven from a selection of even more basic assumptions (which need not explicitly mention associativity), albeit a very clever selection.