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Theorem necon4bid 2716
Description: Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)
Hypothesis
Ref Expression
necon4bid.1  |-  ( ph  ->  ( A  =/=  B  <->  C  =/=  D ) )
Assertion
Ref Expression
necon4bid  |-  ( ph  ->  ( A  =  B  <-> 
C  =  D ) )

Proof of Theorem necon4bid
StepHypRef Expression
1 necon4bid.1 . . 3  |-  ( ph  ->  ( A  =/=  B  <->  C  =/=  D ) )
21necon2bbid 2713 . 2  |-  ( ph  ->  ( C  =  D  <->  -.  A  =/=  B
) )
3 nne 2658 . 2  |-  ( -.  A  =/=  B  <->  A  =  B )
42, 3syl6rbb 262 1  |-  ( ph  ->  ( A  =  B  <-> 
C  =  D ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1395    =/= wne 2652
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 185  df-ne 2654
This theorem is referenced by:  nebi  2767  znnenlem  13957  rpexp  14273  norm-i  26173  trlid0b  36046
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