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Theorem eqeq12 2201
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 2195 . 2  |-  ( A  =  B  ->  ( A  =  C  <->  B  =  C ) )
2 eqeq2 2198 . 2  |-  ( C  =  D  ->  ( B  =  C  <->  B  =  D ) )
31, 2sylan9bb 462 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  =  C  <-> 
B  =  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1363
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 1457  ax-gen 1459  ax-4 1520  ax-17 1536  ax-ext 2170
This theorem depends on definitions:  df-bi 117  df-cleq 2181
This theorem is referenced by:  eqeq12i  2202  eqeq12d  2203  eqeqan12d  2204  funopg  5264  tfri3  6385  th3qlem1  6654  xpdom2  6848  difinfsnlem  7115  difinfsn  7116  xrlttri3  9814  bcn1  10755  summodc  11408  prodmodc  11603  ringinvnz1ne0  13361
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