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Theorem eqeq12 2068
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 2062 . 2  |-  ( A  =  B  ->  ( A  =  C  <->  B  =  C ) )
2 eqeq2 2065 . 2  |-  ( C  =  D  ->  ( B  =  C  <->  B  =  D ) )
31, 2sylan9bb 443 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  =  C  <-> 
B  =  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416  ax-17 1435  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049
This theorem is referenced by:  eqeq12i  2069  eqeq12d  2070  eqeqan12d  2071  funopg  4962  tfri3  5984  th3qlem1  6239  xpdom2  6336  xrlttri3  8819  bcn1  9626
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