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Theorem sseq1d 3185
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
sseq1d  |-  ( ph  ->  ( A  C_  C  <->  B 
C_  C ) )

Proof of Theorem sseq1d
StepHypRef Expression
1 sseq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 sseq1 3179 . 2  |-  ( A  =  B  ->  ( A  C_  C  <->  B  C_  C
) )
31, 2syl 14 1  |-  ( ph  ->  ( A  C_  C  <->  B 
C_  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353    C_ wss 3130
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 3136  df-ss 3143
This theorem is referenced by:  sseq12d  3187  eqsstrd  3192  snssgOLD  3729  ssiun2s  3931  treq  4108  onsucsssucexmid  4527  funimass1  5294  feq1  5349  sbcfg  5365  fvmptssdm  5601  fvimacnvi  5631  nnsucsssuc  6493  ereq1  6542  elpm2r  6666  fipwssg  6978  nnnninf  7124  ctssexmid  7148  iscnp  13702  iscnp4  13721  cnntr  13728  cnconst2  13736  cnptopresti  13741  cnptoprest  13742  txbas  13761  txcnp  13774  txdis  13780  txdis1cn  13781  blssps  13930  blss  13931  ssblex  13934  blin2  13935  metss2  14001  metrest  14009  metcnp3  14014  cnopnap  14097  limccl  14131  ellimc3apf  14132
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