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Theorem necon1d 2692
Description: Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
necon1d.1  |-  ( ph  ->  ( A  =/=  B  ->  C  =  D ) )
Assertion
Ref Expression
necon1d  |-  ( ph  ->  ( C  =/=  D  ->  A  =  B ) )

Proof of Theorem necon1d
StepHypRef Expression
1 necon1d.1 . . 3  |-  ( ph  ->  ( A  =/=  B  ->  C  =  D ) )
2 nne 2668 . . 3  |-  ( -.  C  =/=  D  <->  C  =  D )
31, 2syl6ibr 227 . 2  |-  ( ph  ->  ( A  =/=  B  ->  -.  C  =/=  D
) )
43necon4ad 2687 1  |-  ( ph  ->  ( C  =/=  D  ->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1379    =/= wne 2662
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 2664
This theorem is referenced by:  disji  4440  mul02lem2  9766  xblss2ps  20749  xblss2  20750  lgsne0  23451  h1datomi  26290  eigorthi  26547  disjif  27230  lineintmo  29702  2llnmat  34613  2lnat  34873  tendospcanN  36113  dihmeetlem13N  36409  dochkrshp  36476
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