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Theorem 3sstr4g 3041
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 3026 . 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 1285    C_ wss 2974
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987
This theorem is referenced by:  rabss2  3078  unss2  3144  sslin  3199  ssopab2  4038  xpss12  4473  coss1  4519  coss2  4520  cnvss  4536  rnss  4592  ssres  4665  ssres2  4666  imass1  4730  imass2  4731  imadif  5010  imain  5012  ssoprab2  5592  suppssfv  5739  suppssov1  5740  tposss  5895
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