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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 
|- ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) )

Proof

Step Hyp Ref Expression
1 pm5.501 
 |-  ( ph -> ( ps <-> ( ph <-> ps ) ) )
2 1 bibi1d 
 |-  ( ph -> ( ( ps <-> ch ) <-> ( ( ph <-> ps ) <-> ch ) ) )
3 pm5.501 
 |-  ( ph -> ( ( ps <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) )
4 2 3 bitr3d 
 |-  ( ph -> ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) )
5 nbbn 
 |-  ( ( -. ps <-> ch ) <-> -. ( ps <-> ch ) )
6 nbn2 
 |-  ( -. ph -> ( -. ps <-> ( ph <-> ps ) ) )
7 6 bibi1d 
 |-  ( -. ph -> ( ( -. ps <-> ch ) <-> ( ( ph <-> ps ) <-> ch ) ) )
8 5 7 bitr3id 
 |-  ( -. ph -> ( -. ( ps <-> ch ) <-> ( ( ph <-> ps ) <-> ch ) ) )
9 nbn2 
 |-  ( -. ph -> ( -. ( ps <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) )
10 8 9 bitr3d 
 |-  ( -. ph -> ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) )
11 4 10 pm2.61i 
 |-  ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) )