I asked a question the other day on how to form logical equivalence between a sentence $\phi$ and two other sentences $\psi$ and $\chi$, such that neither $\psi$ nor $\chi$ were on their own as 'strong' as $\phi$. Eventually the answer was provided in the form:
$\vdash\phi\leftrightarrow{(\psi\wedge\chi)}$, and $\not\vdash\chi\rightarrow{\phi}$ and $\not\vdash\psi\rightarrow{\phi}$
Now, I know how to prove provability of the first part semidecidably. I can prove $\vdash\phi\leftrightarrow{(\psi\wedge\chi)}$ by just following proof calculus if $\phi\leftrightarrow{(\psi\wedge\chi)}$ is a decidable sentence. I expect that $\not\vdash\chi\rightarrow{\phi}$ or generally $\not\vdash\phi$ can't be proved for the general case, because then it would seem that would solve the halting problem. But there are methods to prove $\not\vdash\phi$ for specific sentences $\phi$ (or in specific models of a specific theory?) or we wouldn't have independence proofs.
What are some techniques that are used to prove a sentence is undecidable and is there any way to prove $\vdash\phi\leftrightarrow{(\psi\wedge\chi)}$, $\not\vdash\chi\rightarrow{\phi}$ and $\not\vdash\psi\rightarrow{\phi}$ specifically, at least for some instances or models?