I am currently reading these notes on model theory of valued fields, in the section 3.3 appears this theorem:
Theorem. Let $K$ and $L$ be valued fields, with residue fields $k_K$ and $k_L$ respectively, and value groups $\Gamma_K$, $\Gamma_L$ respectively. Assume they satisfy one of the following set of conditions:
(a) The residue fields of $K$ and $L$ are of characteristic $0$.
(b) $K$ and $L$ are of characteristic $0$, the residue fields are of characteristic $p > 0$, the value groups have a smallest positive element, and in both fields the value of $p$ is a fininite multiple $e$ of this smallest positive element.
Then: $(1)$. $$K\equiv L \Longleftrightarrow k_K\equiv k_L \ and \ \Gamma_K \equiv \Gamma_L $$
(2) If $K$ is a valued subfield of $L$, then $$ K\prec L \Longleftrightarrow k_K\prec k_L \ and \ \Gamma_K \prec\Gamma_L$$
this is known as the AKE-principle, but I have not been able to deduce a demonstration of this (in the notes be omitted), any hint is appreciated. Thanks
The proof of the AKE-principle (this stands for Ax-Kochen-Ershov, by the way) is quite involved. You can find a complete proof in these notes by Lou van den Dries (Theorems 5.1 and 6.13).
By the way, Chatzidakis seems to have omitted the hypothesis that $K$ and $L$ are Henselian in her statement of the theorem that you've quoted. She points out in the previous paragraph that it's necessary though!