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