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Theorem necon4i 2506
Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
necon4i.1  |-  ( A  =/=  B  ->  C  =/=  D )
Assertion
Ref Expression
necon4i  |-  ( C  =  D  ->  A  =  B )

Proof of Theorem necon4i
StepHypRef Expression
1 necon4i.1 . . 3  |-  ( A  =/=  B  ->  C  =/=  D )
21necon2bi 2492 . 2  |-  ( C  =  D  ->  -.  A  =/=  B )
3 nne 2450 . 2  |-  ( -.  A  =/=  B  <->  A  =  B )
42, 3sylib 188 1  |-  ( C  =  D  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    =/= wne 2446
This theorem is referenced by:  unixp0  5206  scott0  7556  nn0opthi  11285
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 2448
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