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Theorem sseq1d 3184
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 3178 . 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 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:  sseq12d  3186  eqsstrd  3191  snssgOLD  3727  ssiun2s  3928  treq  4104  onsucsssucexmid  4522  funimass1  5288  feq1  5343  sbcfg  5359  fvmptssdm  5595  fvimacnvi  5625  nnsucsssuc  6486  ereq1  6535  elpm2r  6659  fipwssg  6971  nnnninf  7117  ctssexmid  7141  iscnp  13332  iscnp4  13351  cnntr  13358  cnconst2  13366  cnptopresti  13371  cnptoprest  13372  txbas  13391  txcnp  13404  txdis  13410  txdis1cn  13411  blssps  13560  blss  13561  ssblex  13564  blin2  13565  metss2  13631  metrest  13639  metcnp3  13644  cnopnap  13727  limccl  13761  ellimc3apf  13762
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