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Theorem nfdm 4606
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 4383 . 2 dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
2 nfcv 2194 . . . . 5 𝑥𝑦
3 nfrn.1 . . . . 5 𝑥𝐴
4 nfcv 2194 . . . . 5 𝑥𝑧
52, 3, 4nfbr 3836 . . . 4 𝑥 𝑦𝐴𝑧
65nfex 1544 . . 3 𝑥𝑧 𝑦𝐴𝑧
76nfab 2198 . 2 𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
81, 7nfcxfr 2191 1 𝑥dom 𝐴
Colors of variables: wff set class
Syntax hints:  wex 1397  {cab 2042  wnfc 2181   class class class wbr 3792  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-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-dm 4383
This theorem is referenced by:  nfrn  4607  dmiin  4608  nffn  5023
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