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Theorem necon1d 2668
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 2644 . . 3  |-  ( -.  C  =/=  D  <->  C  =  D )
31, 2syl6ibr 227 . 2  |-  ( ph  ->  ( A  =/=  B  ->  -.  C  =/=  D
) )
43necon4ad 2663 1  |-  ( ph  ->  ( C  =/=  D  ->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1383    =/= wne 2638
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 2640
This theorem is referenced by:  disji  4425  mul02lem2  9760  xblss2ps  20777  xblss2  20778  lgsne0  23480  h1datomi  26371  eigorthi  26628  disjif  27311  lineintmo  29782  2llnmat  34988  2lnat  35248  tendospcanN  36490  dihmeetlem13N  36786  dochkrshp  36853
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