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Theorem sseqtrd 3065
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
sseqtrd.1  |-  ( ph  ->  A  C_  B )
sseqtrd.2  |-  ( ph  ->  B  =  C )
Assertion
Ref Expression
sseqtrd  |-  ( ph  ->  A  C_  C )

Proof of Theorem sseqtrd
StepHypRef Expression
1 sseqtrd.1 . 2  |-  ( ph  ->  A  C_  B )
2 sseqtrd.2 . . 3  |-  ( ph  ->  B  =  C )
32sseq2d 3057 . 2  |-  ( ph  ->  ( A  C_  B  <->  A 
C_  C ) )
41, 3mpbid 146 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1290    C_ wss 3002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-in 3008  df-ss 3015
This theorem is referenced by:  sseqtr4d  3066  fssdmd  5189  resasplitss  5205  nnaword2  6289  erssxp  6331  phpm  6637  ioodisj  9473  tgcl  11827  basgen  11843  bastop1  11846  bastop2  11847  clsss2  11892  topssnei  11925  nninfalllemn  12201
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