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Theorem dmeqd 4565
Description: Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
Hypothesis
Ref Expression
dmeqd.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
dmeqd  |-  ( ph  ->  dom  A  =  dom  B )

Proof of Theorem dmeqd
StepHypRef Expression
1 dmeqd.1 . 2  |-  ( ph  ->  A  =  B )
2 dmeq 4563 . 2  |-  ( A  =  B  ->  dom  A  =  dom  B )
31, 2syl 14 1  |-  ( ph  ->  dom  A  =  dom  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   dom cdm 4371
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  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-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-dm 4381
This theorem is referenced by:  rneq  4589  dmsnsnsng  4828  elxp4  4838  fndmin  5306  1stvalg  5800  fo1st  5815  f1stres  5817  errn  6194  xpassen  6374  xpdom2  6375  frecuzrdgtclt  9503  shftdm  9848
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