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Proof was for soundness, not inconsistency. Anonymous <a href="https://ncatlab.org/nlab/revision/diff/L%C3%B6b%27s+theorem/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/L%C3%B6b%27s+theorem/2">v2</a>, <a href="https://ncatlab.org/nlab/show/L%C3%B6b%27s+theorem">current</a>
Proof was for soundness, not inconsistency.
Anonymous
diff, v2, current
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