RSA derivation of ed=1 mod(ϕ(n))

Question

I know that $m =m^{ed} \bmod n$, and there is Euler's $a^{\varphi(n)} = 1 \bmod n$, but how do we derive that $ed=1 \bmod(\varphi(n))$ holds for all m. Where does that equality come from? Couldn't find a proof for that.