MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nebi Structured version   Visualization version   Unicode version
Theorem nebi 2874
Description: Contraposition law for inequality. (Contributed by NM, 28-Dec-2008.)
Assertion
Ref Expression
nebi |- ( ( A = B <-> C = D ) <-> ( A =/= B <-> C =/= D ) )
Proof of Theorem nebi
StepHypRef Expression
1 id 22 . . 3 |- ( ( A = B <-> C = D ) -> ( A = B <-> C = D ) )
21necon3bid 2838 . 2 |- ( ( A = B <-> C = D ) -> ( A =/= B <-> C =/= D ) )
3 id 22 . . 3 |- ( ( A =/= B <-> C =/= D ) -> ( A =/= B <-> C =/= D ) )
43necon4bid 2839 . 2 |- ( ( A =/= B <-> C =/= D ) -> ( A = B <-> C = D ) )
52, 4impbii 199 1 |- ( ( A = B <-> C = D ) <-> ( A =/= B <-> C =/= D ) )
Colors of variables: wff setvar class
Syntax hints:   <-> wb 196   = wceq 1483   =/= wne 2794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-ne 2795
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator