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Theorem pm5.54dc 865
Description: A conjunction is equivalent to one of its conjuncts, given a decidable conjunct. Based on theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.)
Assertion
Ref Expression
pm5.54dc  |-  (DECID  ph  ->  ( ( ( ph  /\  ps )  <->  ph )  \/  (
( ph  /\  ps )  <->  ps ) ) )

Proof of Theorem pm5.54dc
StepHypRef Expression
1 df-dc 781 . . 3  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 simpr 108 . . . . 5  |-  ( (
ph  /\  ps )  ->  ps )
3 ax-ia3 106 . . . . 5  |-  ( ph  ->  ( ps  ->  ( ph  /\  ps ) ) )
42, 3impbid2 141 . . . 4  |-  ( ph  ->  ( ( ph  /\  ps )  <->  ps ) )
5 simpl 107 . . . . 5  |-  ( (
ph  /\  ps )  ->  ph )
6 ax-in2 580 . . . . 5  |-  ( -. 
ph  ->  ( ph  ->  (
ph  /\  ps )
) )
75, 6impbid2 141 . . . 4  |-  ( -. 
ph  ->  ( ( ph  /\ 
ps )  <->  ph ) )
84, 7orim12i 711 . . 3  |-  ( (
ph  \/  -.  ph )  ->  ( ( ( ph  /\ 
ps )  <->  ps )  \/  ( ( ph  /\  ps )  <->  ph ) ) )
91, 8sylbi 119 . 2  |-  (DECID  ph  ->  ( ( ( ph  /\  ps )  <->  ps )  \/  (
( ph  /\  ps )  <->  ph ) ) )
109orcomd 683 1  |-  (DECID  ph  ->  ( ( ( ph  /\  ps )  <->  ph )  \/  (
( ph  /\  ps )  <->  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> 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-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by: (None)
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