Theorem birot

Description: Rotation of the arguments of the nested implication ( . <-> ( . <-> . ) ) (a general phenomenon for a commutative associative binary operation, see e.g., inrot ) . (Contributed by BJ, 10-Aug-2026)

	Ref	Expression
Assertion	birot	$$⊢\left(\phi \leftrightarrow \left(\psi \leftrightarrow \chi \right)\right)\leftrightarrow \left(\psi \leftrightarrow \left(\chi \leftrightarrow \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bicom	$$⊢\left(\phi \leftrightarrow \left(\psi \leftrightarrow \chi \right)\right)\leftrightarrow \left(\left(\psi \leftrightarrow \chi \right)\leftrightarrow \phi \right)$$
2		biass	$$⊢\left(\left(\psi \leftrightarrow \chi \right)\leftrightarrow \phi \right)\leftrightarrow \left(\psi \leftrightarrow \left(\chi \leftrightarrow \phi \right)\right)$$
3	1 2	bitri	$$⊢\left(\phi \leftrightarrow \left(\psi \leftrightarrow \chi \right)\right)\leftrightarrow \left(\psi \leftrightarrow \left(\chi \leftrightarrow \phi \right)\right)$$