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Theorem pm5.7dc 898
Description: Disjunction distributes over the biconditional, for a decidable proposition. Based on theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbididc 897. (Contributed by Jim Kingdon, 2-Apr-2018.)
Assertion
Ref Expression
pm5.7dc  |-  (DECID  ch  ->  ( ( ( ph  \/  ch )  <->  ( ps  \/  ch ) )  <->  ( ch  \/  ( ph  <->  ps )
) ) )

Proof of Theorem pm5.7dc
StepHypRef Expression
1 orbididc 897 . 2  |-  (DECID  ch  ->  ( ( ch  \/  ( ph 
<->  ps ) )  <->  ( ( ch  \/  ph )  <->  ( ch  \/  ps ) ) ) )
2 orcom 680 . . 3  |-  ( ( ch  \/  ph )  <->  (
ph  \/  ch )
)
3 orcom 680 . . 3  |-  ( ( ch  \/  ps )  <->  ( ps  \/  ch )
)
42, 3bibi12i 227 . 2  |-  ( ( ( ch  \/  ph ) 
<->  ( ch  \/  ps ) )  <->  ( ( ph  \/  ch )  <->  ( ps  \/  ch ) ) )
51, 4syl6rbb 195 1  |-  (DECID  ch  ->  ( ( ( ph  \/  ch )  <->  ( ps  \/  ch ) )  <->  ( ch  \/  ( ph  <->  ps )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    \/ wo 662  DECID wdc 778
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 779
This theorem is referenced by: (None)
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