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Theorem nfdm 4679
Description: Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypothesis
Ref Expression
nfrn.1 𝑥𝐴
Assertion
Ref Expression
nfdm 𝑥dom 𝐴

Proof of Theorem nfdm
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dm 4448 . 2 dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
2 nfcv 2228 . . . . 5 𝑥𝑦
3 nfrn.1 . . . . 5 𝑥𝐴
4 nfcv 2228 . . . . 5 𝑥𝑧
52, 3, 4nfbr 3889 . . . 4 𝑥 𝑦𝐴𝑧
65nfex 1573 . . 3 𝑥𝑧 𝑦𝐴𝑧
76nfab 2233 . 2 𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
81, 7nfcxfr 2225 1 𝑥dom 𝐴
Colors of variables: wff set class
Syntax hints:  wex 1426  {cab 2074  wnfc 2215   class class class wbr 3845  dom cdm 4438
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-dm 4448
This theorem is referenced by:  nfrn  4680  dmiin  4681  nffn  5110
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