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Theorem 3sstr4g 3391
Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
3sstr4g.1  |-  ( ph  ->  A  C_  B )
3sstr4g.2  |-  C  =  A
3sstr4g.3  |-  D  =  B
Assertion
Ref Expression
3sstr4g  |-  ( ph  ->  C  C_  D )

Proof of Theorem 3sstr4g
StepHypRef Expression
1 3sstr4g.1 . 2  |-  ( ph  ->  A  C_  B )
2 3sstr4g.2 . . 3  |-  C  =  A
3 3sstr4g.3 . . 3  |-  D  =  B
42, 3sseq12i 3376 . 2  |-  ( C 
C_  D  <->  A  C_  B
)
51, 4sylibr 205 1  |-  ( ph  ->  C  C_  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    C_ wss 3322
This theorem is referenced by:  rabss2  3428  unss2  3520  sslin  3569  ssopab2  4482  xpss12  4983  coss1  5030  coss2  5031  cnvss  5047  rnss  5100  ssres  5174  ssres2  5175  imass1  5241  imass2  5242  ssoprab2  6132  suppssfv  6303  suppssov1  6304  tposss  6482  onovuni  6606  ss2ixp  7077  fodomfi  7387  cantnfp1lem3  7638  isumsplit  12622  isumrpcl  12625  cvgrat  12662  gsumzf1o  15521  gsumzmhm  15535  gsumzinv  15542  divstgpopn  18151  metnrmlem2  18892  ovolsslem  19382  uniioombllem3  19479  ulmres  20306  xrlimcnp  20809  pntlemq  21297  subgornss  21896  sspba  22228  shlej2i  22883  chpssati  23868  sitgclbn  24659  subfacp1lem6  24873  predpredss  25447  aomclem4  27134  dsmmsubg  27188  bnj1408  29467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-in 3329  df-ss 3336
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