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Theorem 3sstr4g 3550
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 3535 . 2  |-  ( C 
C_  D  <->  A  C_  B
)
51, 4sylibr 212 1  |-  ( ph  ->  C  C_  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = 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:  rabss2  3588  unss2  3680  sslin  3729  ssopab2  4778  xpss12  5113  coss1  5163  coss2  5164  cnvss  5180  rnss  5236  ssres  5304  ssres2  5305  imass1  5376  imass2  5377  ssoprab2  6347  suppssfvOLD  6525  suppssov1OLD  6526  ressuppss  6929  tposss  6966  onovuni  7023  ss2ixp  7492  fodomfi  7809  cantnfp1lem3OLD  8135  isumsplit  13627  isumrpcl  13630  cvgrat  13667  gsumzf1o  16767  gsumzf1oOLD  16770  gsumzmhm  16807  gsumzmhmOLD  16808  gsumzinv  16819  gsumzinvOLD  16820  dsmmsubg  18620  qustgpopn  20463  metnrmlem2  21209  ovolsslem  21740  uniioombllem3  21839  ulmres  22627  xrlimcnp  23141  pntlemq  23629  subgornss  25099  sspba  25431  shlej2i  26088  chpssati  27073  sitgclbn  28078  subfacp1lem6  28422  mthmpps  28735  predpredss  29145  aomclem4  30899  bnj1408  33464  coss12d  37072
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