Propositional interpolation ($A⊢C⊢B$)

If $A⊢B$, eliminate the letters private to $A$ byEvery assignment satisfying $A$ satisfies one disjunct of $C$. If an assignment satisfies $C$, change only the eliminated letters to its witnessing assignment. The modified assignment satisfies $A$ and hence $B$, whose letters were unchanged. Thus $A\models C\models B$, and the propositional completeness theorem gives the two derivations. The interpolant uses only common letters; an empty common vocabulary is handled by logical truth constants.