Prove that $2+2=4$.

I don't know about more concise, but I will sketch the outline of a simpler proof starting with Peano's Axioms for the natural numbers $(N,0,S)$.

We will construct the function $add: N^2\to N$ such that

$\forall a\in N: add(a,0)=a$

$\forall a,b\in N: add(a,S(b))=S(add(a,b))$

Begin by constructing the set $add$ such that

$\forall a,b,c:[(a,b,c)\in add \iff (a,b,c)\in N^3$

$\land \forall d:[\forall e,f,g: [(e,f,g)\in d\implies (e,f,g)\in N^3]$

$\land \forall e\in N: (e,0,e)\in d$

$\land \forall e,f,g:[(e,f,g)\in d \implies (e,S(f),S(g))\in d]$

$\implies (a,b,c)\in d]]$

Then prove that $add$ is the required function (see full formal proof in DC Proof format, 728 lines).

Then define $1=S(0), 2=S(1), 3=S(2), 4=S(3).$

Then prove, in turn, that $add(2,0)=2, add(2,1)=3, add(2,2)=4$ as required.