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Theorem nebi 2530
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 19 . . 3  |-  ( ( A  =  B  <->  C  =  D )  ->  ( A  =  B  <->  C  =  D ) )
21necon3bid 2494 . 2  |-  ( ( A  =  B  <->  C  =  D )  ->  ( A  =/=  B  <->  C  =/=  D ) )
3 id 19 . . 3  |-  ( ( A  =/=  B  <->  C  =/=  D )  ->  ( A  =/=  B  <->  C  =/=  D
) )
43necon4bid 2525 . 2  |-  ( ( A  =/=  B  <->  C  =/=  D )  ->  ( A  =  B  <->  C  =  D
) )
52, 4impbii 180 1  |-  ( ( A  =  B  <->  C  =  D )  <->  ( A  =/=  B  <->  C  =/=  D
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    =/= wne 2459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-ne 2461
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