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Theorem sseq12 3033
Description: Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
Assertion
Ref Expression
sseq12  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  C_  C  <->  B 
C_  D ) )

Proof of Theorem sseq12
StepHypRef Expression
1 sseq1 3031 . 2  |-  ( A  =  B  ->  ( A  C_  C  <->  B  C_  C
) )
2 sseq2 3032 . 2  |-  ( C  =  D  ->  ( B  C_  C  <->  B  C_  D
) )
31, 2sylan9bb 450 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  C_  C  <->  B 
C_  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    C_ wss 2984
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-in 2990  df-ss 2997
This theorem is referenced by:  sseq12i  3036  undifexmid  3992  exmidundif  3999  funcnvuni  5036  fun11iun  5222
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