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Proposition: For all formulas A and B, the following holds:
A <=> B iff B <=> A A <=> B iff (A /\ B) \/ (~A /\ ~B) A <=> B iff (A => B) /\ (B => A)
Equivalence can thus be defined by implication and conjunction.
Proposition: Equivalence is not associative.
Proof: by construction of a counterexample.