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Theorem 3netr3d 2472
Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
Hypotheses
Ref Expression
3netr3d.1  |-  ( ph  ->  A  =/=  B )
3netr3d.2  |-  ( ph  ->  A  =  C )
3netr3d.3  |-  ( ph  ->  B  =  D )
Assertion
Ref Expression
3netr3d  |-  ( ph  ->  C  =/=  D )

Proof of Theorem 3netr3d
StepHypRef Expression
1 3netr3d.1 . 2  |-  ( ph  ->  A  =/=  B )
2 3netr3d.2 . . 3  |-  ( ph  ->  A  =  C )
3 3netr3d.3 . . 3  |-  ( ph  ->  B  =  D )
42, 3neeq12d 2461 . 2  |-  ( ph  ->  ( A  =/=  B  <->  C  =/=  D ) )
51, 4mpbid 201 1  |-  ( ph  ->  C  =/=  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    =/= wne 2446
This theorem is referenced by:  subrgnzr  16019  dchrisum0re  20662  pellex  26920  cdlemg9a  30821  cdlemg11aq  30827  cdlemg12b  30833  cdlemg12  30839  cdlemg13  30841  cdlemg19  30873  cdlemk3  31022  cdlemk12  31039  cdlemk12u  31061  lclkrlem2g  31703  mapdncol  31860  mapdpglem29  31890  hdmaprnlem1N  32042  hdmap14lem9  32069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-cleq 2276  df-ne 2448
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