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Theorem pm13.181 2409
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 2159 . 2  |-  ( A  =  B  <->  B  =  A )
2 pm13.18 2408 . 2  |-  ( ( B  =  A  /\  B  =/=  C )  ->  A  =/=  C )
31, 2sylanb 282 1  |-  ( ( A  =  B  /\  B  =/=  C )  ->  A  =/=  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    =/= wne 2327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1427  ax-gen 1429  ax-4 1490  ax-17 1506  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-cleq 2150  df-ne 2328
This theorem is referenced by:  nninfisollemne  7074  fzprval  9984  mod2eq1n2dvds  11769
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