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Theorem biass 348
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
StepHypRef Expression
1 pm5.501 330 . . . 4  |-  ( ph  ->  ( ps  <->  ( ph  <->  ps ) ) )
21bibi1d 310 . . 3  |-  ( ph  ->  ( ( ps  <->  ch )  <->  ( ( ph  <->  ps )  <->  ch ) ) )
3 pm5.501 330 . . 3  |-  ( ph  ->  ( ( ps  <->  ch )  <->  (
ph 
<->  ( ps  <->  ch )
) ) )
42, 3bitr3d 246 . 2  |-  ( ph  ->  ( ( ( ph  <->  ps )  <->  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) ) )
5 nbbn 347 . . . 4  |-  ( ( -.  ps  <->  ch )  <->  -.  ( ps  <->  ch )
)
6 nbn2 334 . . . . 5  |-  ( -. 
ph  ->  ( -.  ps  <->  (
ph 
<->  ps ) ) )
76bibi1d 310 . . . 4  |-  ( -. 
ph  ->  ( ( -. 
ps 
<->  ch )  <->  ( ( ph 
<->  ps )  <->  ch )
) )
85, 7syl5bbr 250 . . 3  |-  ( -. 
ph  ->  ( -.  ( ps 
<->  ch )  <->  ( ( ph 
<->  ps )  <->  ch )
) )
9 nbn2 334 . . 3  |-  ( -. 
ph  ->  ( -.  ( ps 
<->  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) ) )
108, 9bitr3d 246 . 2  |-  ( -. 
ph  ->  ( ( (
ph 
<->  ps )  <->  ch )  <->  (
ph 
<->  ( ps  <->  ch )
) ) )
114, 10pm2.61i 156 1  |-  ( ( ( ph  <->  ps )  <->  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176
This theorem is referenced by:  biluk  899  xorass  1299  had1  1392  symdifass  24441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177
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