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Theorem nfdm 4783
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 4549 . 2 dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
2 nfcv 2281 . . . . 5 𝑥𝑦
3 nfrn.1 . . . . 5 𝑥𝐴
4 nfcv 2281 . . . . 5 𝑥𝑧
52, 3, 4nfbr 3974 . . . 4 𝑥 𝑦𝐴𝑧
65nfex 1616 . . 3 𝑥𝑧 𝑦𝐴𝑧
76nfab 2286 . 2 𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
81, 7nfcxfr 2278 1 𝑥dom 𝐴
Colors of variables: wff set class
Syntax hints:  wex 1468  {cab 2125  wnfc 2268   class class class wbr 3929  dom cdm 4539
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-dm 4549
This theorem is referenced by:  nfrn  4784  dmiin  4785  nffn  5219  ellimc3apf  12808
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