I was trying to find the precise and strongest statement of the Downward Lowenheim-Skolem-Tarski theorem for Cofinality logic $\mathcal{L}(Q^{cf}_\lambda)$- first order logic enhanced with the quantifier $Q^{cf}_\lambda$, where
$M\vDash Q^{cf}_\lambda xy\phi(x,y)$ $\iff$ $\{(a,b)\in M^2\mid M\vDash\phi(a,b)\}$ is a linear order of cofinality $\lambda$
The reference I found was to Shelah's Generalized quantifiers and compact logic where I found the following very general theorem:
who's proof uses elements from the proof of a previous theorem (of compactness for a certain cofinality logic).
First, I want to understand what this gives for the specific case of just $Q^{cf}_\lambda$. If we set $n=0, \mu=1$ then we get:
If $M$ is an $L$-model of cardinality $>\kappa$ where $\kappa\geq\lambda,|L|$, and there is some regular $\lambda'\leq \kappa$ with $\lambda'\ne\lambda$, then $M$ has an $\mathcal{L}(Q^{cf}_\lambda)$-elementary submodel of power $\kappa$.
In particular, for $\lambda=\omega$: If $M$ is an $L$-model of cardinality $>\kappa$ where $\kappa\geq\aleph_1,|L|$, then $M$ has an $\mathcal{L}(Q^{cf}_\omega)$-elementary submodel of power $\kappa$.
Is that correct?
Is there a reference for a self-contained proof of only the more specific case, or even just the case of cofinality $\omega$?