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Theorem necon1d 2677
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 2654 . . 3  |-  ( -.  C  =/=  D  <->  C  =  D )
31, 2syl6ibr 227 . 2  |-  ( ph  ->  ( A  =/=  B  ->  -.  C  =/=  D
) )
43necon4ad 2672 1  |-  ( ph  ->  ( C  =/=  D  ->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1370    =/= wne 2648
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 2650
This theorem is referenced by:  disji  4391  mul02lem2  9661  xblss2ps  20118  xblss2  20119  lgsne0  22815  h1datomi  25163  eigorthi  25420  disjif  26100  lineintmo  28355  2llnmat  33531  2lnat  33791  tendospcanN  35031  dihmeetlem13N  35327  dochkrshp  35394
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