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Theorem pm5.71dc 908
Description: Decidable proposition version of theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.) (Modified for decidability by Jim Kingdon, 19-Apr-2018.)
Assertion
Ref Expression
pm5.71dc  |-  (DECID  ps  ->  ( ( ps  ->  -.  ch )  ->  ( ( ( ph  \/  ps )  /\  ch )  <->  ( ph  /\ 
ch ) ) ) )

Proof of Theorem pm5.71dc
StepHypRef Expression
1 orel2 681 . . . . 5  |-  ( -. 
ps  ->  ( ( ph  \/  ps )  ->  ph )
)
2 orc 669 . . . . 5  |-  ( ph  ->  ( ph  \/  ps ) )
31, 2impbid1 141 . . . 4  |-  ( -. 
ps  ->  ( ( ph  \/  ps )  <->  ph ) )
43anbi1d 454 . . 3  |-  ( -. 
ps  ->  ( ( (
ph  \/  ps )  /\  ch )  <->  ( ph  /\ 
ch ) ) )
54a1i 9 . 2  |-  (DECID  ps  ->  ( -.  ps  ->  (
( ( ph  \/  ps )  /\  ch )  <->  (
ph  /\  ch )
) ) )
6 pm2.21 583 . . 3  |-  ( -. 
ch  ->  ( ch  ->  ( ( ph  \/  ps ) 
<-> 
ph ) ) )
76pm5.32rd 440 . 2  |-  ( -. 
ch  ->  ( ( (
ph  \/  ps )  /\  ch )  <->  ( ph  /\ 
ch ) ) )
85, 7jadc 799 1  |-  (DECID  ps  ->  ( ( ps  ->  -.  ch )  ->  ( ( ( ph  \/  ps )  /\  ch )  <->  ( ph  /\ 
ch ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 665  DECID wdc 781
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 581  ax-io 666
This theorem depends on definitions:  df-bi 116  df-dc 782
This theorem is referenced by: (None)
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