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Theorem bj-dcbi 11702
Description: Equivalence property for DECID. TODO: solve conflict with dcbi 882; minimize dcbii 785 and dcbid 786 with it, as well as theorems using those. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-dcbi  |-  ( (
ph 
<->  ps )  ->  (DECID  ph  <-> DECID  ps ) )

Proof of Theorem bj-dcbi
StepHypRef Expression
1 id 19 . . 3  |-  ( (
ph 
<->  ps )  ->  ( ph 
<->  ps ) )
2 bj-notbi 11699 . . 3  |-  ( (
ph 
<->  ps )  ->  ( -.  ph  <->  -.  ps )
)
31, 2orbi12d 742 . 2  |-  ( (
ph 
<->  ps )  ->  (
( ph  \/  -.  ph )  <->  ( ps  \/  -.  ps ) ) )
4 df-dc 781 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
5 df-dc 781 . 2  |-  (DECID  ps  <->  ( ps  \/  -.  ps ) )
63, 4, 53bitr4g 221 1  |-  ( (
ph 
<->  ps )  ->  (DECID  ph  <-> DECID  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    \/ wo 664  DECID wdc 780
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 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by:  bj-d0clsepcl  11703
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