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Theorem 3sstr3d 3236
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 3223 . 2  |-  ( ph  ->  ( A  C_  B  <->  C 
C_  D ) )
51, 4mpbid 147 1  |-  ( ph  ->  C  C_  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1372    C_ wss 3165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186
This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-in 3171  df-ss 3178
This theorem is referenced by:  fnsnsplitss  5782
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