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Theorem biassdc 1327
 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 DECID DECID

Proof of Theorem biassdc
StepHypRef Expression
1 df-dc 777 . . 3 DECID
2 pm5.501 242 . . . . . . 7
32bibi1d 231 . . . . . 6
4 pm5.501 242 . . . . . 6
53, 4bitr3d 188 . . . . 5
65a1d 22 . . . 4 DECID DECID
7 nbbndc 1326 . . . . . . . . 9 DECID DECID
87imp 122 . . . . . . . 8 DECID DECID
98adantl 271 . . . . . . 7 DECID DECID
10 nbn2 646 . . . . . . . . 9
1110bibi1d 231 . . . . . . . 8
1211adantr 270 . . . . . . 7 DECID DECID
139, 12bitr3d 188 . . . . . 6 DECID DECID
14 nbn2 646 . . . . . . 7
1514adantr 270 . . . . . 6 DECID DECID
1613, 15bitr3d 188 . . . . 5 DECID DECID
1716ex 113 . . . 4 DECID DECID
186, 17jaoi 669 . . 3 DECID DECID
191, 18sylbi 119 . 2 DECID DECID DECID
2019expd 254 1 DECID DECID DECID
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 102   wb 103   wo 662  DECID wdc 776 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663 This theorem depends on definitions:  df-bi 115  df-dc 777 This theorem is referenced by:  bilukdc  1328
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