ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  con2biddc Unicode version

Theorem con2biddc 880
Description: A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.)
Hypothesis
Ref Expression
con2biddc.1  |-  ( ph  ->  (DECID  ch  ->  ( ps  <->  -. 
ch ) ) )
Assertion
Ref Expression
con2biddc  |-  ( ph  ->  (DECID  ch  ->  ( ch  <->  -. 
ps ) ) )

Proof of Theorem con2biddc
StepHypRef Expression
1 con2biddc.1 . . . 4  |-  ( ph  ->  (DECID  ch  ->  ( ps  <->  -. 
ch ) ) )
2 bicom 140 . . . 4  |-  ( ( ps  <->  -.  ch )  <->  ( -.  ch  <->  ps )
)
31, 2imbitrdi 161 . . 3  |-  ( ph  ->  (DECID  ch  ->  ( -.  ch 
<->  ps ) ) )
43con1biddc 876 . 2  |-  ( ph  ->  (DECID  ch  ->  ( -.  ps 
<->  ch ) ) )
5 bicom 140 . 2  |-  ( ( -.  ps  <->  ch )  <->  ( ch  <->  -.  ps )
)
64, 5imbitrdi 161 1  |-  ( ph  ->  (DECID  ch  ->  ( ch  <->  -. 
ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 105  DECID wdc 834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835
This theorem is referenced by:  anordc  956  xor3dc  1387
  Copyright terms: Public domain W3C validator