ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  necon4bbiddc Unicode version

Theorem necon4bbiddc 2329
Description: Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
Hypothesis
Ref Expression
necon4bbiddc.1  |-  ( ph  ->  (DECID  ps  ->  (DECID  A  =  B  ->  ( -.  ps  <->  A  =/=  B ) ) ) )
Assertion
Ref Expression
necon4bbiddc  |-  ( ph  ->  (DECID  ps  ->  (DECID  A  =  B  ->  ( ps  <->  A  =  B ) ) ) )

Proof of Theorem necon4bbiddc
StepHypRef Expression
1 necon4bbiddc.1 . . . . . 6  |-  ( ph  ->  (DECID  ps  ->  (DECID  A  =  B  ->  ( -.  ps  <->  A  =/=  B ) ) ) )
2 bicom 138 . . . . . 6  |-  ( ( -.  ps  <->  A  =/=  B )  <->  ( A  =/= 
B  <->  -.  ps )
)
31, 2syl8ib 164 . . . . 5  |-  ( ph  ->  (DECID  ps  ->  (DECID  A  =  B  ->  ( A  =/= 
B  <->  -.  ps )
) ) )
43com23 77 . . . 4  |-  ( ph  ->  (DECID  A  =  B  -> 
(DECID 
ps  ->  ( A  =/= 
B  <->  -.  ps )
) ) )
54necon4abiddc 2328 . . 3  |-  ( ph  ->  (DECID  A  =  B  -> 
(DECID 
ps  ->  ( A  =  B  <->  ps ) ) ) )
65com23 77 . 2  |-  ( ph  ->  (DECID  ps  ->  (DECID  A  =  B  ->  ( A  =  B  <->  ps ) ) ) )
7 bicom 138 . 2  |-  ( ( A  =  B  <->  ps )  <->  ( ps  <->  A  =  B
) )
86, 7syl8ib 164 1  |-  ( ph  ->  (DECID  ps  ->  (DECID  A  =  B  ->  ( ps  <->  A  =  B ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103  DECID wdc 780    = wceq 1289    =/= wne 2255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781  df-ne 2256
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator