For two proposition logical formulas $\phi$ and $\chi$ so that $\phi\implies\chi$ is generally valid. How can I prove that there is a formular $\psi$ with $var(\psi )\subseteq var(\phi )\cap var(\chi )$ so that $\phi\implies\psi$ and $\psi\implies\chi$ are generally valid?
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The formula $\psi$ is called an interpolant and the method of finding such an interpolant is called Craig interpolation. On the Wikipedia article you can find more information about it, including a proof that interpolants always exist in propositional logic.
