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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:

$[[x]]_I$	$[[\neg x]]_I$	$[[x \Leftrightarrow \neg x]]_I$
$1$	$0$	$0$
$0$	$1$	$0$

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

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References

Bibliography

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