If a consistent propositional theory locally omits every member of a countable family of propositional types, it has a Boolean valuation omitting them all. One may also impose any finite condition consistent with the theory. Successively extend the condition by a negated member of the next type; local omission preserves consistency. The propositional compactness theorem then supplies the valuation. Countability of the ambient language and decidability of consistency are not needed for this argument.
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