MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm13.181 Unicode version

Theorem pm13.181 2532
Description: Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
Assertion
Ref Expression
pm13.181  |-  ( ( A  =  B  /\  B  =/=  C )  ->  A  =/=  C )

Proof of Theorem pm13.181
StepHypRef Expression
1 eqcom 2298 . 2  |-  ( A  =  B  <->  B  =  A )
2 pm13.18 2531 . 2  |-  ( ( B  =  A  /\  B  =/=  C )  ->  A  =/=  C )
31, 2sylanb 458 1  |-  ( ( A  =  B  /\  B  =/=  C )  ->  A  =/=  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    =/= wne 2459
This theorem is referenced by:  fzprval  10860  hdrmp  25809  a9e2ndeqALT  29024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-cleq 2289  df-ne 2461
  Copyright terms: Public domain W3C validator