ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  necon1addc Unicode version

Theorem necon1addc 2410
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
Hypothesis
Ref Expression
necon1addc.1  |-  ( ph  ->  (DECID  ps  ->  ( -.  ps  ->  A  =  B ) ) )
Assertion
Ref Expression
necon1addc  |-  ( ph  ->  (DECID  ps  ->  ( A  =/=  B  ->  ps )
) )

Proof of Theorem necon1addc
StepHypRef Expression
1 df-ne 2335 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
2 necon1addc.1 . . 3  |-  ( ph  ->  (DECID  ps  ->  ( -.  ps  ->  A  =  B ) ) )
3 con1dc 846 . . 3  |-  (DECID  ps  ->  ( ( -.  ps  ->  A  =  B )  -> 
( -.  A  =  B  ->  ps )
) )
42, 3sylcom 28 . 2  |-  ( ph  ->  (DECID  ps  ->  ( -.  A  =  B  ->  ps ) ) )
51, 4syl7bi 164 1  |-  ( ph  ->  (DECID  ps  ->  ( A  =/=  B  ->  ps )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4  DECID wdc 824    = wceq 1342    =/= wne 2334
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 821  df-dc 825  df-ne 2335
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator