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Theorem eqtr 2175
Description: Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.)
Assertion
Ref Expression
eqtr  |-  ( ( A  =  B  /\  B  =  C )  ->  A  =  C )

Proof of Theorem eqtr
StepHypRef Expression
1 eqeq1 2164 . 2  |-  ( A  =  B  ->  ( A  =  C  <->  B  =  C ) )
21biimpar 295 1  |-  ( ( A  =  B  /\  B  =  C )  ->  A  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335
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-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
This theorem is referenced by:  eqtr2  2176  eqtr3  2177  sylan9eq  2210  eqvinc  2835  eqvincg  2836  uneqdifeqim  3479  preqsn  3738  dtruex  4518  relresfld  5115  relcoi1  5117  eqer  6512  xpider  6551  addlsub  8245  bj-findis  13554
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