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Theorem eqsstr3d 3059
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
eqsstr3d.1  |-  ( ph  ->  B  =  A )
eqsstr3d.2  |-  ( ph  ->  B  C_  C )
Assertion
Ref Expression
eqsstr3d  |-  ( ph  ->  A  C_  C )

Proof of Theorem eqsstr3d
StepHypRef Expression
1 eqsstr3d.1 . . 3  |-  ( ph  ->  B  =  A )
21eqcomd 2093 . 2  |-  ( ph  ->  A  =  B )
3 eqsstr3d.2 . 2  |-  ( ph  ->  B  C_  C )
42, 3eqsstrd 3058 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    C_ wss 2997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3003  df-ss 3010
This theorem is referenced by:  ssxpbm  4853  ssxp1  4854  ssxp2  4855  suppssof1  5854  tfrlemiubacc  6077  tfr1onlemubacc  6093  tfrcllemubacc  6106  oaword1  6214  phplem4dom  6558  fisseneq  6621  archnqq  6955  nnsf  11541
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