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Theorem eqtr2 2196
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 2179 . 2  |-  ( A  =  B  <->  B  =  A )
2 eqtr 2195 . 2  |-  ( ( B  =  A  /\  A  =  C )  ->  B  =  C )
31, 2sylanb 284 1  |-  ( ( A  =  B  /\  A  =  C )  ->  B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-cleq 2170
This theorem is referenced by:  eqvinc  2861  eqvincg  2862  moop2  4252  reusv3i  4460  relop  4778  f0rn0  5411  fliftfun  5797  th3qlem1  6637  enq0ref  7432  enq0tr  7433  genpdisj  7522  addlsub  8327  fsum2dlemstep  11442  0dvds  11818  cncongr1  12103
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