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Theorem sseqtrd 3045
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 3037 . 2  |-  ( ph  ->  ( A  C_  B  <->  A 
C_  C ) )
41, 3mpbid 145 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    C_ wss 2983
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-in 2989  df-ss 2996
This theorem is referenced by:  sseqtr4d  3046  resasplitss  5121  nnaword2  6175  erssxp  6217  phpm  6422  ioodisj  9127
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