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Theorem pm13.18 2639
Description: Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
Assertion
Ref Expression
pm13.18  |-  ( ( A  =  B  /\  A  =/=  C )  ->  B  =/=  C )

Proof of Theorem pm13.18
StepHypRef Expression
1 eqeq1 2410 . . . 4  |-  ( A  =  B  ->  ( A  =  C  <->  B  =  C ) )
21biimprd 215 . . 3  |-  ( A  =  B  ->  ( B  =  C  ->  A  =  C ) )
32necon3d 2605 . 2  |-  ( A  =  B  ->  ( A  =/=  C  ->  B  =/=  C ) )
43imp 419 1  |-  ( ( A  =  B  /\  A  =/=  C )  ->  B  =/=  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    =/= wne 2567
This theorem is referenced by:  pm13.181  2640  4atexlemex4  30555
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-11 1757  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-cleq 2397  df-ne 2569
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