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Theorem sseq12 3532
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 3530 . 2  |-  ( A  =  B  ->  ( A  C_  C  <->  B  C_  C
) )
2 sseq2 3531 . 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 1379    C_ wss 3481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-in 3488  df-ss 3495
This theorem is referenced by:  sseq12i  3535  sorpsscmpl  6586  funcnvuni  6748  fun11iun  6755  sornom  8669  axdc3lem2  8843  ipole  15662  ipodrsima  15669  cmetss  21621  funpsstri  29122  ismrcd2  30559  ismrc  30561
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