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Theorem orbididc 895
Description: Disjunction distributes over the biconditional, for a decidable proposition. Based on an axiom of system DS in Vladimir Lifschitz, "On calculational proofs" (1998), http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.3384. (Contributed by Jim Kingdon, 2-Apr-2018.)
Assertion
Ref Expression
orbididc  |-  (DECID  ph  ->  ( ( ph  \/  ( ps 
<->  ch ) )  <->  ( ( ph  \/  ps )  <->  ( ph  \/  ch ) ) ) )

Proof of Theorem orbididc
StepHypRef Expression
1 orimdidc 846 . . 3  |-  (DECID  ph  ->  ( ( ph  \/  ( ps  ->  ch ) )  <-> 
( ( ph  \/  ps )  ->  ( ph  \/  ch ) ) ) )
2 orimdidc 846 . . 3  |-  (DECID  ph  ->  ( ( ph  \/  ( ch  ->  ps ) )  <-> 
( ( ph  \/  ch )  ->  ( ph  \/  ps ) ) ) )
31, 2anbi12d 457 . 2  |-  (DECID  ph  ->  ( ( ( ph  \/  ( ps  ->  ch )
)  /\  ( ph  \/  ( ch  ->  ps ) ) )  <->  ( (
( ph  \/  ps )  ->  ( ph  \/  ch ) )  /\  (
( ph  \/  ch )  ->  ( ph  \/  ps ) ) ) ) )
4 dfbi2 380 . . . 4  |-  ( ( ps  <->  ch )  <->  ( ( ps  ->  ch )  /\  ( ch  ->  ps )
) )
54orbi2i 712 . . 3  |-  ( (
ph  \/  ( ps  <->  ch ) )  <->  ( ph  \/  ( ( ps  ->  ch )  /\  ( ch 
->  ps ) ) ) )
6 ordi 763 . . 3  |-  ( (
ph  \/  ( ( ps  ->  ch )  /\  ( ch  ->  ps )
) )  <->  ( ( ph  \/  ( ps  ->  ch ) )  /\  ( ph  \/  ( ch  ->  ps ) ) ) )
75, 6bitri 182 . 2  |-  ( (
ph  \/  ( ps  <->  ch ) )  <->  ( ( ph  \/  ( ps  ->  ch ) )  /\  ( ph  \/  ( ch  ->  ps ) ) ) )
8 dfbi2 380 . 2  |-  ( ( ( ph  \/  ps ) 
<->  ( ph  \/  ch ) )  <->  ( (
( ph  \/  ps )  ->  ( ph  \/  ch ) )  /\  (
( ph  \/  ch )  ->  ( ph  \/  ps ) ) ) )
93, 7, 83bitr4g 221 1  |-  (DECID  ph  ->  ( ( ph  \/  ( ps 
<->  ch ) )  <->  ( ( ph  \/  ps )  <->  ( ph  \/  ch ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> 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-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by:  pm5.7dc  896
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