Theorem biass 374

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 in Principia 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.)

Assertion

Ref	Expression
biass	|- ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) )

Proof of Theorem biass

Step	Hyp	Ref	Expression
1		pm5.501 356	. . . 4 |- ( ph -> ( ps <-> ( ph <-> ps ) ) )
2	1	bibi1d 333	. . 3 |- ( ph -> ( ( ps <-> ch ) <-> ( ( ph <-> ps ) <-> ch ) ) )
3		pm5.501 356	. . 3 |- ( ph -> ( ( ps <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) )
4	2, 3	bitr3d 270	. 2 |- ( ph -> ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) )
5		nbbn 373	. . . 4 |- ( ( -. ps <-> ch ) <-> -. ( ps <-> ch ) )
6		nbn2 360	. . . . 5 |- ( -. ph -> ( -. ps <-> ( ph <-> ps ) ) )
7	6	bibi1d 333	. . . 4 |- ( -. ph -> ( ( -. ps <-> ch ) <-> ( ( ph <-> ps ) <-> ch ) ) )
8	5, 7	syl5bbr 274	. . 3 |- ( -. ph -> ( -. ( ps <-> ch ) <-> ( ( ph <-> ps ) <-> ch ) ) )
9		nbn2 360	. . 3 |- ( -. ph -> ( -. ( ps <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) )
10	8, 9	bitr3d 270	. 2 |- ( -. ph -> ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) )
11	4, 10	pm2.61i 176	1 |- ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) )

Syntax hints:   -. wn 3   <-> wb 196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197

This theorem is referenced by:  biluk  974  xorass  1468  had1  1542  symdifass  3853