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Proof

(related to Lemma: A proposition cannot be equivalent to its negation)

• Let $x$ be a proposition.
• Using the semantics of propositional logic and the definition of equivalence we get the following truth table:

• It follows that \(x\Leftrightarrow \neg x\) is a contradiction.
• Thus, $x$ cannot be equivalent to its negation.
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References

Bibliography

1. Mendelson Elliott: "Theory and Problems of Boolean Algebra and Switching Circuits", McGraw-Hill Book Company, 1982