The semantic version of the deduction theorem says that $\{A_{1}, …, A_{n}\}\vDash$ B $\iff \vDash A_{1}, …, A_{n} \to$ B. I read that the deduction theorem does not hold in modal logic, but upon looking deeper they always only seem to talk about the syntactic version ($\vdash$) of the deduction theorem.
So does the semantic deduction theorem hold in Modal Logic?
I am able to answer my own question. First I found a YT-video where a professor of philosophy specifically talks about the semantic version of the deduction theorem, proving it for propositional logic and then confirms that it also holds for Modal Logic (of course just relative to a model and a certain world $w_{i}$, so for instance for a T-model it would look like this: $V(w_{i})\{A_{1}, …, A_{n}\}\vDash_{T}$ $V(w_{i})$B $\iff \vDash_{T} V(w_{i})$$(A_{1}, …, A_{n} \to$ B). Here is the link: https://www.youtube.com/watch?v=WuLCL2q8Lx4
I even asked him specifically in the comment section, so here is what he wrote:
Me: So the semantic deduction theorem, i.e. {A1, …, An} |= B iff |= A1, …, An -> B, does also hold in Modal Logic (of course not globally but just for a particular model)?
Him: Yes, it holds in modal logic, because propositional logic holds at each world. So anywhere where the premises are true, their conjunction is also true, and if B follows, then any world will support the corresponding material implication.
Here is a more formal proof from someone with more knowledge than mine that looks spot on:
$\begin{aligned}(\Gamma\cup\{A\}\models B) &\iff \forall M\colon\forall w\in W(M)\colon (M,w\models\Gamma\cup\{A\})\to (M,w\models B)\\ &\iff \forall M\colon\forall w\in W(M)\colon (M,w\models\Gamma)\land (M,w\models A)\to (M,w\models B)\\ &\iff \forall M\colon\forall w\in W(M)\colon (M,w\models\Gamma)\to ((M,w\models A)\to (M,w\models B))\\ &\iff \forall M\colon\forall w\in W(M)\colon (M,w\models\Gamma)\to (M,w\models A\to B)\\ &\iff (\Gamma\models A\to B)\end{aligned}$
All what I wrote refers to the notion of the local semantic consequence where we look at a certain world to check if there is a semantic consequence in this certain world. In a notion of a global semantic consequence the semantic deduction theorem might not hold, that is another topic.
Feel free to add things I missed or might be of educational value.