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Theorem eqeq12 2128
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 2122 . 2  |-  ( A  =  B  ->  ( A  =  C  <->  B  =  C ) )
2 eqeq2 2125 . 2  |-  ( C  =  D  ->  ( B  =  C  <->  B  =  D ) )
31, 2sylan9bb 455 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  =  C  <-> 
B  =  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314
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 1406  ax-gen 1408  ax-4 1470  ax-17 1489  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-cleq 2108
This theorem is referenced by:  eqeq12i  2129  eqeq12d  2130  eqeqan12d  2131  funopg  5125  tfri3  6230  th3qlem1  6497  xpdom2  6691  difinfsnlem  6950  difinfsn  6951  xrlttri3  9523  bcn1  10444  summodc  11092
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