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Theorem 3sstr3d 3220
Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
Hypotheses
Ref Expression
3sstr3d.1  |-  ( ph  ->  A  C_  B )
3sstr3d.2  |-  ( ph  ->  A  =  C )
3sstr3d.3  |-  ( ph  ->  B  =  D )
Assertion
Ref Expression
3sstr3d  |-  ( ph  ->  C  C_  D )

Proof of Theorem 3sstr3d
StepHypRef Expression
1 3sstr3d.1 . 2  |-  ( ph  ->  A  C_  B )
2 3sstr3d.2 . . 3  |-  ( ph  ->  A  =  C )
3 3sstr3d.3 . . 3  |-  ( ph  ->  B  =  D )
42, 3sseq12d 3207 . 2  |-  ( ph  ->  ( A  C_  B  <->  C 
C_  D ) )
51, 4mpbid 201 1  |-  ( ph  ->  C  C_  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    C_ wss 3152
This theorem is referenced by:  cnvtsr  14331  dprdss  15264  dprd2da  15277  dmdprdsplit2lem  15280  mplind  16243  txcmplem1  17335  setsmstopn  18024  tngtopn  18166  bcthlem2  18747  bcthlem4  18749  uniiccvol  18935  dyadmaxlem  18952  dvlip2  19342  dvne0  19358  shlej2  21940  bnd2lem  26515  heiborlem8  26542  hbtlem5  27332  dochord  31560  lclkrlem2p  31712  mapdsn  31831
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166
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