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Theorem 3sstr3g 3040
Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
Hypotheses
Ref Expression
3sstr3g.1  |-  ( ph  ->  A  C_  B )
3sstr3g.2  |-  A  =  C
3sstr3g.3  |-  B  =  D
Assertion
Ref Expression
3sstr3g  |-  ( ph  ->  C  C_  D )

Proof of Theorem 3sstr3g
StepHypRef Expression
1 3sstr3g.1 . 2  |-  ( ph  ->  A  C_  B )
2 3sstr3g.2 . . 3  |-  A  =  C
3 3sstr3g.3 . . 3  |-  B  =  D
42, 3sseq12i 3026 . 2  |-  ( A 
C_  B  <->  C  C_  D
)
51, 4sylib 120 1  |-  ( ph  ->  C  C_  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    C_ wss 2974
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 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987
This theorem is referenced by: (None)
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