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Theorem eqtr 2099
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 2088 . 2  |-  ( A  =  B  ->  ( A  =  C  <->  B  =  C ) )
21biimpar 291 1  |-  ( ( A  =  B  /\  B  =  C )  ->  A  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-4 1441  ax-17 1460  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-cleq 2075
This theorem is referenced by:  eqtr2  2100  eqtr3  2101  sylan9eq  2134  eqvinc  2719  eqvincg  2720  uneqdifeqim  3335  preqsn  3575  dtruex  4310  relresfld  4877  relcoi1  4879  eqer  6204  xpiderm  6243  addlsub  7541  bj-findis  10932
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