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Theorem 3sstr3d 2987
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 2974 . 2  |-  ( ph  ->  ( A  C_  B  <->  C 
C_  D ) )
51, 4mpbid 135 1  |-  ( ph  ->  C  C_  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    C_ wss 2917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931
This theorem is referenced by: (None)
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