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Theorem 3sstr3g 3390
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 3376 . 2  |-  ( A 
C_  B  <->  C  C_  D
)
51, 4sylib 190 1  |-  ( ph  ->  C  C_  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    C_ wss 3322
This theorem is referenced by:  uniintsn  4089  fpwwe2lem13  8519  hmeocls  17802  hmeontr  17803  chsscon3i  22965  pjss1coi  23668  mdslmd2i  23835  ssbnd  26499  bnd2lem  26502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-in 3329  df-ss 3336
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