Theorem biass 348

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		pm5.501 330	. . . 4 ⊢ (φ → (ψ ↔ (φ ↔ ψ)))
2	1	bibi1d 310	. . 3 ⊢ (φ → ((ψ ↔ χ) ↔ ((φ ↔ ψ) ↔ χ)))
3		pm5.501 330	. . 3 ⊢ (φ → ((ψ ↔ χ) ↔ (φ ↔ (ψ ↔ χ))))
4	2, 3	bitr3d 246	. 2 ⊢ (φ → (((φ ↔ ψ) ↔ χ) ↔ (φ ↔ (ψ ↔ χ))))
5		nbbn 347	. . . 4 ⊢ ((¬ ψ ↔ χ) ↔ ¬ (ψ ↔ χ))
6		nbn2 334	. . . . 5 ⊢ (¬ φ → (¬ ψ ↔ (φ ↔ ψ)))
7	6	bibi1d 310	. . . 4 ⊢ (¬ φ → ((¬ ψ ↔ χ) ↔ ((φ ↔ ψ) ↔ χ)))
8	5, 7	syl5bbr 250	. . 3 ⊢ (¬ φ → (¬ (ψ ↔ χ) ↔ ((φ ↔ ψ) ↔ χ)))
9		nbn2 334	. . 3 ⊢ (¬ φ → (¬ (ψ ↔ χ) ↔ (φ ↔ (ψ ↔ χ))))
10	8, 9	bitr3d 246	. 2 ⊢ (¬ φ → (((φ ↔ ψ) ↔ χ) ↔ (φ ↔ (ψ ↔ χ))))
11	4, 10	pm2.61i 156	1 ⊢ (((φ ↔ ψ) ↔ χ) ↔ (φ ↔ (ψ ↔ χ)))

Syntax hints:  ¬ wn 3   ↔ wb 176

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 177

This theorem is referenced by:  biluk  899  xorass  1308  had1  1402