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Theorem dmeqd 4537
 Description: Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
Hypothesis
Ref Expression
dmeqd.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
dmeqd (𝜑 → dom 𝐴 = dom 𝐵)

Proof of Theorem dmeqd
StepHypRef Expression
1 dmeqd.1 . 2 (𝜑𝐴 = 𝐵)
2 dmeq 4535 . 2 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
31, 2syl 14 1 (𝜑 → dom 𝐴 = dom 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243  dom cdm 4345 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-dm 4355 This theorem is referenced by:  rneq  4561  dmsnsnsng  4798  elxp4  4808  fndmin  5274  1stvalg  5769  fo1st  5784  f1stres  5786  errn  6128  xpassen  6304  xpdom2  6305  shftdm  9423
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