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Theorem 3sstr3g 3529
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 3515 . 2  |-  ( A 
C_  B  <->  C  C_  D
)
51, 4sylib 196 1  |-  ( ph  ->  C  C_  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    C_ wss 3461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-in 3468  df-ss 3475
This theorem is referenced by:  uniintsn  4309  fpwwe2lem13  9023  hmeocls  20247  hmeontr  20248  chsscon3i  26357  pjss1coi  27060  mdslmd2i  27227  ssbnd  30260  bnd2lem  30263  nzss  31198
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