Theorem biass 374

Description: Associative law for the biconditional. An axiom of system DS in Vladimir Lifschitz, "On calculational proofs", Annals of Pure and Applied Logic, 113:207-224, 2002, http://www.cs.utexas.edu/users/ai-lab/pub-view.php?PubID=26805. Interestingly, this law was not included in Principia Mathematica but was apparently first noted by Jan Lukasiewicz circa 1923. (Contributed by NM, 8-Jan-2005.) (Proof shortened by Juha Arpiainen, 19-Jan-2006.) (Proof shortened by Wolf Lammen, 21-Sep-2013.)

Assertion

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

Proof of Theorem biass

Step	Hyp	Ref	Expression
1		pm5.501 356	. . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓)))
2	1	bibi1d 333	. . 3 ⊢ (𝜑 → ((𝜓 ↔ 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ 𝜒)))
3		pm5.501 356	. . 3 ⊢ (𝜑 → ((𝜓 ↔ 𝜒) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒))))
4	2, 3	bitr3d 270	. 2 ⊢ (𝜑 → (((𝜑 ↔ 𝜓) ↔ 𝜒) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒))))
5		nbbn 373	. . . 4 ⊢ ((¬ 𝜓 ↔ 𝜒) ↔ ¬ (𝜓 ↔ 𝜒))
6		nbn2 360	. . . . 5 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜑 ↔ 𝜓)))
7	6	bibi1d 333	. . . 4 ⊢ (¬ 𝜑 → ((¬ 𝜓 ↔ 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ 𝜒)))
8	5, 7	syl5bbr 274	. . 3 ⊢ (¬ 𝜑 → (¬ (𝜓 ↔ 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ 𝜒)))
9		nbn2 360	. . 3 ⊢ (¬ 𝜑 → (¬ (𝜓 ↔ 𝜒) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒))))
10	8, 9	bitr3d 270	. 2 ⊢ (¬ 𝜑 → (((𝜑 ↔ 𝜓) ↔ 𝜒) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒))))
11	4, 10	pm2.61i 176	1 ⊢ (((𝜑 ↔ 𝜓) ↔ 𝜒) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒)))

Syntax hints:  ¬ wn 3   ↔ wb 196

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 197

This theorem is referenced by:  biluk  974  xorass  1468  had1  1542  symdifass  3853