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Theorem eqtr2 2100
Description: A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
eqtr2  |-  ( ( A  =  B  /\  A  =  C )  ->  B  =  C )

Proof of Theorem eqtr2
StepHypRef Expression
1 eqcom 2084 . 2  |-  ( A  =  B  <->  B  =  A )
2 eqtr 2099 . 2  |-  ( ( B  =  A  /\  A  =  C )  ->  B  =  C )
31, 2sylanb 278 1  |-  ( ( A  =  B  /\  A  =  C )  ->  B  =  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:  eqvinc  2719  eqvincg  2720  moop2  4014  reusv3i  4217  relop  4514  fliftfun  5467  th3qlem1  6274  enq0ref  6685  enq0tr  6686  genpdisj  6775  addlsub  7541  0dvds  10360  cncongr1  10629
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