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Theorem con34bdc 839
Description: Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116, but for a decidable proposition. (Contributed by Jim Kingdon, 24-Apr-2018.)
Assertion
Ref Expression
con34bdc  |-  (DECID  ps  ->  ( ( ph  ->  ps ) 
<->  ( -.  ps  ->  -. 
ph ) ) )

Proof of Theorem con34bdc
StepHypRef Expression
1 con3 614 . 2  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )
2 condc 821 . 2  |-  (DECID  ps  ->  ( ( -.  ps  ->  -. 
ph )  ->  ( ph  ->  ps ) ) )
31, 2impbid2 142 1  |-  (DECID  ps  ->  ( ( ph  ->  ps ) 
<->  ( -.  ps  ->  -. 
ph ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104  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-in1 586  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803
This theorem is referenced by:  pm4.14dc  858  algcvgblem  11626
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