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	⊢ ( ( 𝜑 ↔ ( 𝜓 ↔ 𝜒 ) ) ↔ ( 𝜓 ↔ ( 𝜒 ↔ 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bicom	⊢ ( ( 𝜑 ↔ ( 𝜓 ↔ 𝜒 ) ) ↔ ( ( 𝜓 ↔ 𝜒 ) ↔ 𝜑 ) )
2		biass	⊢ ( ( ( 𝜓 ↔ 𝜒 ) ↔ 𝜑 ) ↔ ( 𝜓 ↔ ( 𝜒 ↔ 𝜑 ) ) )
3	1 2	bitri	⊢ ( ( 𝜑 ↔ ( 𝜓 ↔ 𝜒 ) ) ↔ ( 𝜓 ↔ ( 𝜒 ↔ 𝜑 ) ) )