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Theorem pm5.55dc 881
Description: A disjunction is equivalent to one of its disjuncts, given a decidable disjunct. Based on theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.)
Assertion
Ref Expression
pm5.55dc  |-  (DECID  ph  ->  ( ( ( ph  \/  ps )  <->  ph )  \/  (
( ph  \/  ps ) 
<->  ps ) ) )

Proof of Theorem pm5.55dc
StepHypRef Expression
1 df-dc 803 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 biort 797 . . . 4  |-  ( ph  ->  ( ph  <->  ( ph  \/  ps ) ) )
32bicomd 140 . . 3  |-  ( ph  ->  ( ( ph  \/  ps )  <->  ph ) )
4 biorf 716 . . . 4  |-  ( -. 
ph  ->  ( ps  <->  ( ph  \/  ps ) ) )
54bicomd 140 . . 3  |-  ( -. 
ph  ->  ( ( ph  \/  ps )  <->  ps )
)
63, 5orim12i 731 . 2  |-  ( (
ph  \/  -.  ph )  ->  ( ( ( ph  \/  ps )  <->  ph )  \/  ( ( ph  \/  ps )  <->  ps ) ) )
71, 6sylbi 120 1  |-  (DECID  ph  ->  ( ( ( ph  \/  ps )  <->  ph )  \/  (
( ph  \/  ps ) 
<->  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    \/ wo 680  DECID wdc 802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-dc 803
This theorem is referenced by: (None)
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