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Theorem nfdm 4753
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  |-  F/_ x A
Assertion
Ref Expression
nfdm  |-  F/_ x dom  A

Proof of Theorem nfdm
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dm 4519 . 2  |-  dom  A  =  { y  |  E. z  y A z }
2 nfcv 2258 . . . . 5  |-  F/_ x
y
3 nfrn.1 . . . . 5  |-  F/_ x A
4 nfcv 2258 . . . . 5  |-  F/_ x
z
52, 3, 4nfbr 3944 . . . 4  |-  F/ x  y A z
65nfex 1601 . . 3  |-  F/ x E. z  y A
z
76nfab 2263 . 2  |-  F/_ x { y  |  E. z  y A z }
81, 7nfcxfr 2255 1  |-  F/_ x dom  A
Colors of variables: wff set class
Syntax hints:   E.wex 1453   {cab 2103   F/_wnfc 2245   class class class wbr 3899   dom cdm 4509
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-dm 4519
This theorem is referenced by:  nfrn  4754  dmiin  4755  nffn  5189  ellimc3apf  12725
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