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Theorem pm4.14dc 876
Description: Theorem *4.14 of [WhiteheadRussell] p. 117, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.)
Assertion
Ref Expression
pm4.14dc  |-  (DECID  ch  ->  ( ( ( ph  /\  ps )  ->  ch )  <->  ( ( ph  /\  -.  ch )  ->  -.  ps ) ) )

Proof of Theorem pm4.14dc
StepHypRef Expression
1 con34bdc 857 . . 3  |-  (DECID  ch  ->  ( ( ps  ->  ch ) 
<->  ( -.  ch  ->  -. 
ps ) ) )
21imbi2d 229 . 2  |-  (DECID  ch  ->  ( ( ph  ->  ( ps  ->  ch ) )  <-> 
( ph  ->  ( -. 
ch  ->  -.  ps )
) ) )
3 impexp 261 . 2  |-  ( ( ( ph  /\  ps )  ->  ch )  <->  ( ph  ->  ( ps  ->  ch ) ) )
4 impexp 261 . 2  |-  ( ( ( ph  /\  -.  ch )  ->  -.  ps ) 
<->  ( ph  ->  ( -.  ch  ->  -.  ps )
) )
52, 3, 43bitr4g 222 1  |-  (DECID  ch  ->  ( ( ( ph  /\  ps )  ->  ch )  <->  ( ( ph  /\  -.  ch )  ->  -.  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104  DECID wdc 820
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-in1 604  ax-in2 605  ax-io 699
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821
This theorem is referenced by: (None)
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