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Theorem sseq2i 3524
Description: An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
Hypothesis
Ref Expression
sseq1i.1  |-  A  =  B
Assertion
Ref Expression
sseq2i  |-  ( C 
C_  A  <->  C  C_  B
)

Proof of Theorem sseq2i
StepHypRef Expression
1 sseq1i.1 . 2  |-  A  =  B
2 sseq2 3521 . 2  |-  ( A  =  B  ->  ( C  C_  A  <->  C  C_  B
) )
31, 2ax-mp 5 1  |-  ( C 
C_  A  <->  C  C_  B
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1395    C_ wss 3471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-in 3478  df-ss 3485
This theorem is referenced by:  sseqtri  3531  syl6sseq  3545  abss  3565  ssrab  3574  ssindif0  3883  difcom  3915  ssunsn2  4191  ssunpr  4194  sspr  4195  sstp  4196  ssintrab  4312  iunpwss  4425  dffun2  5604  ssimaex  5938  elpwun  6612  frfi  7783  alephislim  8481  cardaleph  8487  fin1a2lem12  8808  zornn0g  8902  ssxr  9671  nnwo  11172  isstruct  14654  issubm  16105  grpissubg  16348  islinds  18971  basdif0  19581  tgdif0  19621  cmpsublem  20026  cmpsub  20027  hauscmplem  20033  2ndcctbss  20082  fbncp  20466  cnextfval  20688  eltsms  20757  reconn  21459  axcontlem3  24396  axcontlem4  24397  chsscon1i  26507  hatomistici  27408  chirredlem4  27439  atabs2i  27448  mdsymlem1  27449  mdsymlem3  27451  mdsymlem6  27454  mdsymlem8  27456  dmdbr5ati  27468  iundifdif  27569  nocvxminlem  29646  nocvxmin  29647  ismblfin  30239  stoweidlem57  32021  issubmgm  32718  linccl  33138  lincdifsn  33148
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