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Theorem sseq12 3047
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 3045 . 2  |-  ( A  =  B  ->  ( A  C_  C  <->  B  C_  C
) )
2 sseq2 3046 . 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 1289    C_ wss 2997
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-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3003  df-ss 3010
This theorem is referenced by:  sseq12i  3050  undifexmid  4019  exmidundif  4026  funcnvuni  5069  fun11iun  5258
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