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Theorem dmeqd 4565
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 4563 . 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 1259  dom cdm 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-dm 4383
This theorem is referenced by:  rneq  4589  dmsnsnsng  4826  elxp4  4836  fndmin  5302  1stvalg  5797  fo1st  5812  f1stres  5814  errn  6159  xpassen  6335  xpdom2  6336  shftdm  9651
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