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Theorem pm5.54dc 886
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 803 . . 3  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 simpr 109 . . . . 5  |-  ( (
ph  /\  ps )  ->  ps )
3 ax-ia3 107 . . . . 5  |-  ( ph  ->  ( ps  ->  ( ph  /\  ps ) ) )
42, 3impbid2 142 . . . 4  |-  ( ph  ->  ( ( ph  /\  ps )  <->  ps ) )
5 simpl 108 . . . . 5  |-  ( (
ph  /\  ps )  ->  ph )
6 ax-in2 587 . . . . 5  |-  ( -. 
ph  ->  ( ph  ->  (
ph  /\  ps )
) )
75, 6impbid2 142 . . . 4  |-  ( -. 
ph  ->  ( ( ph  /\ 
ps )  <->  ph ) )
84, 7orim12i 731 . . 3  |-  ( (
ph  \/  -.  ph )  ->  ( ( ( ph  /\ 
ps )  <->  ps )  \/  ( ( ph  /\  ps )  <->  ph ) ) )
91, 8sylbi 120 . 2  |-  (DECID  ph  ->  ( ( ( ph  /\  ps )  <->  ps )  \/  (
( ph  /\  ps )  <->  ph ) ) )
109orcomd 701 1  |-  (DECID  ph  ->  ( ( ( ph  /\  ps )  <->  ph )  \/  (
( ph  /\  ps )  <->  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> 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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