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Theorem biassdc 1286
Description: Associative law for the biconditional, for decidable propositions.

The classical version (without the decidability conditions) is 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, and, interestingly, was not included in Principia Mathematica but was apparently first noted by Jan Lukasiewicz circa 1923. (Contributed by Jim Kingdon, 4-May-2018.)

Assertion
Ref Expression
biassdc  |-  (DECID  ph  ->  (DECID  ps 
->  (DECID  ch  ->  ( (
( ph  <->  ps )  <->  ch )  <->  (
ph 
<->  ( ps  <->  ch )
) ) ) ) )

Proof of Theorem biassdc
StepHypRef Expression
1 df-dc 743 . . 3  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 pm5.501 233 . . . . . . 7  |-  ( ph  ->  ( ps  <->  ( ph  <->  ps ) ) )
32bibi1d 222 . . . . . 6  |-  ( ph  ->  ( ( ps  <->  ch )  <->  ( ( ph  <->  ps )  <->  ch ) ) )
4 pm5.501 233 . . . . . 6  |-  ( ph  ->  ( ( ps  <->  ch )  <->  (
ph 
<->  ( ps  <->  ch )
) ) )
53, 4bitr3d 179 . . . . 5  |-  ( ph  ->  ( ( ( ph  <->  ps )  <->  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) ) )
65a1d 22 . . . 4  |-  ( ph  ->  ( (DECID  ps  /\ DECID  ch )  ->  (
( ( ph  <->  ps )  <->  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) ) ) )
7 nbbndc 1285 . . . . . . . . 9  |-  (DECID  ps  ->  (DECID  ch 
->  ( ( -.  ps  <->  ch )  <->  -.  ( ps  <->  ch ) ) ) )
87imp 115 . . . . . . . 8  |-  ( (DECID  ps 
/\ DECID  ch )  ->  ( ( -.  ps  <->  ch )  <->  -.  ( ps  <->  ch )
) )
98adantl 262 . . . . . . 7  |-  ( ( -.  ph  /\  (DECID  ps  /\ DECID  ch ) )  ->  (
( -.  ps  <->  ch )  <->  -.  ( ps  <->  ch )
) )
10 nbn2 613 . . . . . . . . 9  |-  ( -. 
ph  ->  ( -.  ps  <->  (
ph 
<->  ps ) ) )
1110bibi1d 222 . . . . . . . 8  |-  ( -. 
ph  ->  ( ( -. 
ps 
<->  ch )  <->  ( ( ph 
<->  ps )  <->  ch )
) )
1211adantr 261 . . . . . . 7  |-  ( ( -.  ph  /\  (DECID  ps  /\ DECID  ch ) )  ->  (
( -.  ps  <->  ch )  <->  ( ( ph  <->  ps )  <->  ch ) ) )
139, 12bitr3d 179 . . . . . 6  |-  ( ( -.  ph  /\  (DECID  ps  /\ DECID  ch ) )  ->  ( -.  ( ps  <->  ch )  <->  ( ( ph  <->  ps )  <->  ch ) ) )
14 nbn2 613 . . . . . . 7  |-  ( -. 
ph  ->  ( -.  ( ps 
<->  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) ) )
1514adantr 261 . . . . . 6  |-  ( ( -.  ph  /\  (DECID  ps  /\ DECID  ch ) )  ->  ( -.  ( ps  <->  ch )  <->  (
ph 
<->  ( ps  <->  ch )
) ) )
1613, 15bitr3d 179 . . . . 5  |-  ( ( -.  ph  /\  (DECID  ps  /\ DECID  ch ) )  ->  (
( ( ph  <->  ps )  <->  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) ) )
1716ex 108 . . . 4  |-  ( -. 
ph  ->  ( (DECID  ps  /\ DECID  ch )  ->  ( ( ( ph  <->  ps )  <->  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) ) ) )
186, 17jaoi 636 . . 3  |-  ( (
ph  \/  -.  ph )  ->  ( (DECID  ps  /\ DECID  ch )  ->  (
( ( ph  <->  ps )  <->  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) ) ) )
191, 18sylbi 114 . 2  |-  (DECID  ph  ->  ( (DECID  ps  /\ DECID  ch )  ->  (
( ( ph  <->  ps )  <->  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) ) ) )
2019expd 245 1  |-  (DECID  ph  ->  (DECID  ps 
->  (DECID  ch  ->  ( (
( ph  <->  ps )  <->  ch )  <->  (
ph 
<->  ( ps  <->  ch )
) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    <-> wb 98    \/ wo 629  DECID wdc 742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630
This theorem depends on definitions:  df-bi 110  df-dc 743
This theorem is referenced by:  bilukdc  1287
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