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Theorem sseq2d 3127
Description: An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
Hypothesis
Ref Expression
sseq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
sseq2d  |-  ( ph  ->  ( C  C_  A  <->  C 
C_  B ) )

Proof of Theorem sseq2d
StepHypRef Expression
1 sseq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 sseq2 3121 . 2  |-  ( A  =  B  ->  ( C  C_  A  <->  C  C_  B
) )
31, 2syl 14 1  |-  ( ph  ->  ( C  C_  A  <->  C 
C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    C_ wss 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084
This theorem is referenced by:  sseq12d  3128  sseqtrd  3135  exmidsssn  4125  exmidsssnc  4126  onsucsssucexmid  4442  sbcrel  4625  funimass2  5201  fnco  5231  fnssresb  5235  fnimaeq0  5244  foimacnv  5385  fvelimab  5477  ssimaexg  5483  fvmptss2  5496  rdgss  6280  summodclem2  11158  summodc  11159  zsumdc  11160  fsum3cvg3  11172  prodmodclem2  11353  prodmodc  11354  ennnfoneleminc  11931  isbasisg  12221  tgval  12228  tgss3  12257  restbasg  12347  tgrest  12348  restopn2  12362  cnpnei  12398  cnptopresti  12417  txbas  12437  elmopn  12625  neibl  12670  dvfgg  12836
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