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Theorem 3sstr4g 3198
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 3183 . 2  |-  ( C 
C_  D  <->  A  C_  B
)
51, 4sylibr 134 1  |-  ( ph  ->  C  C_  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    C_ wss 3129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142
This theorem is referenced by:  rabss2  3238  unss2  3306  sslin  3361  ssopab2  4271  xpss12  4729  coss1  4777  coss2  4778  cnvss  4795  rnss  4852  ssres  4928  ssres2  4929  imass1  4998  imass2  4999  imadif  5291  imain  5293  ssoprab2  5924  suppssfv  6072  suppssov1  6073  tposss  6240  ss2ixp  6704  isumsplit  11470  isumrpcl  11473
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