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Theorem nfdm 4853
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 4619 . 2 dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
2 nfcv 2312 . . . . 5 𝑥𝑦
3 nfrn.1 . . . . 5 𝑥𝐴
4 nfcv 2312 . . . . 5 𝑥𝑧
52, 3, 4nfbr 4033 . . . 4 𝑥 𝑦𝐴𝑧
65nfex 1630 . . 3 𝑥𝑧 𝑦𝐴𝑧
76nfab 2317 . 2 𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
81, 7nfcxfr 2309 1 𝑥dom 𝐴
Colors of variables: wff set class
Syntax hints:  wex 1485  {cab 2156  wnfc 2299   class class class wbr 3987  dom cdm 4609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-dm 4619
This theorem is referenced by:  nfrn  4854  dmiin  4855  nffn  5292  ellimc3apf  13344
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