Theorem bj-bibibi 33922

Description: A property of the biconditional. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-bibibi	⊢ (𝜑 ↔ (𝜓 ↔ (𝜑 ↔ 𝜓)))

Proof of Theorem bj-bibibi

Step	Hyp	Ref	Expression
1		pm5.501 369	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓)))
2		bianir 1053	. . . 4 ⊢ ((𝜓 ∧ (𝜑 ↔ 𝜓)) → 𝜑)
3	2	ex 415	. . 3 ⊢ (𝜓 → ((𝜑 ↔ 𝜓) → 𝜑))
4		bibif 374	. . . . 5 ⊢ (¬ 𝜓 → ((𝜑 ↔ 𝜓) ↔ ¬ 𝜑))
5	4	con2bid 357	. . . 4 ⊢ (¬ 𝜓 → (𝜑 ↔ ¬ (𝜑 ↔ 𝜓)))
6	5	biimprd 250	. . 3 ⊢ (¬ 𝜓 → (¬ (𝜑 ↔ 𝜓) → 𝜑))
7	3, 6	bija 384	. 2 ⊢ ((𝜓 ↔ (𝜑 ↔ 𝜓)) → 𝜑)
8	1, 7	impbii 211	1 ⊢ (𝜑 ↔ (𝜓 ↔ (𝜑 ↔ 𝜓)))

Syntax hints:  ¬ wn 3   ↔ wb 208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-an 399

This theorem is referenced by: (None)