ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  necon1abiddc Unicode version

Theorem necon1abiddc 2402
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon1abiddc.1  |-  ( ph  ->  (DECID  ps  ->  ( -.  ps 
<->  A  =  B ) ) )
Assertion
Ref Expression
necon1abiddc  |-  ( ph  ->  (DECID  ps  ->  ( A  =/=  B  <->  ps ) ) )

Proof of Theorem necon1abiddc
StepHypRef Expression
1 necon1abiddc.1 . . 3  |-  ( ph  ->  (DECID  ps  ->  ( -.  ps 
<->  A  =  B ) ) )
21con1biddc 871 . 2  |-  ( ph  ->  (DECID  ps  ->  ( -.  A  =  B  <->  ps )
) )
3 df-ne 2341 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
43bibi1i 227 . 2  |-  ( ( A  =/=  B  <->  ps )  <->  ( -.  A  =  B  <->  ps ) )
52, 4syl6ibr 161 1  |-  ( ph  ->  (DECID  ps  ->  ( A  =/=  B  <->  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104  DECID wdc 829    = wceq 1348    =/= wne 2340
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 609  ax-in2 610  ax-io 704
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-ne 2341
This theorem is referenced by:  necon2abiddc  2406
  Copyright terms: Public domain W3C validator