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Metamath Proof Explorer

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 ) ) )