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Theorem pm13.18 2520
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 2291 . . . 4  |-  ( A  =  B  ->  ( A  =  C  <->  B  =  C ) )
21biimprd 214 . . 3  |-  ( A  =  B  ->  ( B  =  C  ->  A  =  C ) )
32necon3d 2486 . 2  |-  ( A  =  B  ->  ( A  =/=  C  ->  B  =/=  C ) )
43imp 418 1  |-  ( ( A  =  B  /\  A  =/=  C )  ->  B  =/=  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    =/= wne 2448
This theorem is referenced by:  pm13.181  2521  4atexlemex4  30335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-11 1717  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-an 360  df-cleq 2278  df-ne 2450
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