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Theorem 3sstr4g 3056
Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
3sstr4g.1  |-  ( ph  ->  A  C_  B )
3sstr4g.2  |-  C  =  A
3sstr4g.3  |-  D  =  B
Assertion
Ref Expression
3sstr4g  |-  ( ph  ->  C  C_  D )

Proof of Theorem 3sstr4g
StepHypRef Expression
1 3sstr4g.1 . 2  |-  ( ph  ->  A  C_  B )
2 3sstr4g.2 . . 3  |-  C  =  A
3 3sstr4g.3 . . 3  |-  D  =  B
42, 3sseq12i 3041 . 2  |-  ( C 
C_  D  <->  A  C_  B
)
51, 4sylibr 132 1  |-  ( ph  ->  C  C_  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287    C_ wss 2988
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-in 2994  df-ss 3001
This theorem is referenced by:  rabss2  3093  unss2  3160  sslin  3215  ssopab2  4075  xpss12  4512  coss1  4558  coss2  4559  cnvss  4576  rnss  4632  ssres  4705  ssres2  4706  imass1  4771  imass2  4772  imadif  5056  imain  5058  ssoprab2  5656  suppssfv  5803  suppssov1  5804  tposss  5959
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