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Theorem 3sstr3g 3231
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 3217 . 2  |-  ( A 
C_  B  <->  C  C_  D
)
51, 4sylib 188 1  |-  ( ph  ->  C  C_  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    C_ wss 3165
This theorem is referenced by:  uniintsn  3915  fpwwe2lem13  8280  hmeocls  17475  hmeontr  17476  chsscon3i  22056  pjss1coi  22759  mdslmd2i  22926  dmrngcmp  25854  ssbnd  26615  bnd2lem  26618
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179
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