Theorem birot 389

Description: Rotation of the arguments of the nested implication (. ↔ (. ↔ .)) (a general phenomenon for a commutative associative binary operation, see e.g., inrot 4188) . (Contributed by BJ, 10-Aug-2026.)

Assertion

Ref	Expression
birot	⊢ ((𝜑 ↔ (𝜓 ↔ 𝜒)) ↔ (𝜓 ↔ (𝜒 ↔ 𝜑)))

Proof of Theorem birot

Step	Hyp	Ref	Expression
1		bicom 225	. 2 ⊢ ((𝜑 ↔ (𝜓 ↔ 𝜒)) ↔ ((𝜓 ↔ 𝜒) ↔ 𝜑))
2		biass 388	. 2 ⊢ (((𝜓 ↔ 𝜒) ↔ 𝜑) ↔ (𝜓 ↔ (𝜒 ↔ 𝜑)))
3	1, 2	bitri 278	1 ⊢ ((𝜑 ↔ (𝜓 ↔ 𝜒)) ↔ (𝜓 ↔ (𝜒 ↔ 𝜑)))

This proof depends on syntax axioms:   ↔ wb 209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by:  had1  1633