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Theorem eqeq12 2100
Description: Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)
Assertion
Ref Expression
eqeq12  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  =  C  <-> 
B  =  D ) )

Proof of Theorem eqeq12
StepHypRef Expression
1 eqeq1 2094 . 2  |-  ( A  =  B  ->  ( A  =  C  <->  B  =  C ) )
2 eqeq2 2097 . 2  |-  ( C  =  D  ->  ( B  =  C  <->  B  =  D ) )
31, 2sylan9bb 450 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  =  C  <-> 
B  =  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289
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 1381  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081
This theorem is referenced by:  eqeq12i  2101  eqeq12d  2102  eqeqan12d  2103  funopg  5048  tfri3  6132  th3qlem1  6394  xpdom2  6547  xrlttri3  9267  bcn1  10166  isummo  10773
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