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Theorem sseq12 3391
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 3389 . 2  |-  ( A  =  B  ->  ( A  C_  C  <->  B  C_  C
) )
2 sseq2 3390 . 2  |-  ( C  =  D  ->  ( B  C_  C  <->  B  C_  D
) )
31, 2sylan9bb 699 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  C_  C  <->  B 
C_  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    C_ wss 3340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-in 3347  df-ss 3354
This theorem is referenced by:  sseq12i  3394  sorpsscmpl  6383  funcnvuni  6542  fun11iun  6549  sornom  8458  axdc3lem2  8632  ipole  15340  ipodrsima  15347  cmetss  20837  funpsstri  27588  ismrcd2  29047  ismrc  29049
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