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