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Proof

(related to Corollary: Commutativity of Equivalence)

• Let $L$ be a set of propositions and let $I$ be an interpretation with the valuation function $[[]]_I$.
• According to the truth table of equivalence, the valuation function $[[x\Leftrightarrow y]]_I$ does not change its value if we exchange the valuations of $[[x]]_I$ and $[[y]]_I,$
• Thus, the equivalence operation "$\Leftrightarrow$" is commutative, i.e. $x \Leftrightarrow y=y \Leftrightarrow x$.
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References

Bibliography

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