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Theorem pm13.181 2302
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 2058 . 2  |-  ( A  =  B  <->  B  =  A )
2 pm13.18 2301 . 2  |-  ( ( B  =  A  /\  B  =/=  C )  ->  A  =/=  C )
31, 2sylanb 272 1  |-  ( ( A  =  B  /\  B  =/=  C )  ->  A  =/=  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    =/= wne 2220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-4 1416  ax-17 1435  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-ne 2221
This theorem is referenced by:  fzprval  9046  mod2eq1n2dvds  10191
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