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Theorem sseq12 3201
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 3199 . 2  |-  ( A  =  B  ->  ( A  C_  C  <->  B  C_  C
) )
2 sseq2 3200 . 2  |-  ( C  =  D  ->  ( B  C_  C  <->  B  C_  D
) )
31, 2sylan9bb 680 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  C_  C  <->  B 
C_  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    C_ wss 3152
This theorem is referenced by:  sseq12i  3204  funcnvuni  5317  fun11iun  5493  sorpsscmpl  6288  sornom  7903  axdc3lem2  8077  ipole  14261  ipodrsima  14268  cmetss  18740  funpsstri  24121  isprsr  25224  ismrcd2  26774  ismrc  26776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166
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