Theorem biass

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 inPrincipia 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)

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

Proof

Step	Hyp	Ref	Expression
1		pm5.501	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜑 ↔ 𝜓 ) ) )
2	1	bibi1d	⊢ ( 𝜑 → ( ( 𝜓 ↔ 𝜒 ) ↔ ( ( 𝜑 ↔ 𝜓 ) ↔ 𝜒 ) ) )
3		pm5.501	⊢ ( 𝜑 → ( ( 𝜓 ↔ 𝜒 ) ↔ ( 𝜑 ↔ ( 𝜓 ↔ 𝜒 ) ) ) )
4	2 3	bitr3d	⊢ ( 𝜑 → ( ( ( 𝜑 ↔ 𝜓 ) ↔ 𝜒 ) ↔ ( 𝜑 ↔ ( 𝜓 ↔ 𝜒 ) ) ) )
5		nbbn	⊢ ( ( ¬ 𝜓 ↔ 𝜒 ) ↔ ¬ ( 𝜓 ↔ 𝜒 ) )
6		nbn2	⊢ ( ¬ 𝜑 → ( ¬ 𝜓 ↔ ( 𝜑 ↔ 𝜓 ) ) )
7	6	bibi1d	⊢ ( ¬ 𝜑 → ( ( ¬ 𝜓 ↔ 𝜒 ) ↔ ( ( 𝜑 ↔ 𝜓 ) ↔ 𝜒 ) ) )
8	5 7	bitr3id	⊢ ( ¬ 𝜑 → ( ¬ ( 𝜓 ↔ 𝜒 ) ↔ ( ( 𝜑 ↔ 𝜓 ) ↔ 𝜒 ) ) )
9		nbn2	⊢ ( ¬ 𝜑 → ( ¬ ( 𝜓 ↔ 𝜒 ) ↔ ( 𝜑 ↔ ( 𝜓 ↔ 𝜒 ) ) ) )
10	8 9	bitr3d	⊢ ( ¬ 𝜑 → ( ( ( 𝜑 ↔ 𝜓 ) ↔ 𝜒 ) ↔ ( 𝜑 ↔ ( 𝜓 ↔ 𝜒 ) ) ) )
11	4 10	pm2.61i	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ↔ 𝜒 ) ↔ ( 𝜑 ↔ ( 𝜓 ↔ 𝜒 ) ) )