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Theorem nfdm 4778
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 4544 . 2  |-  dom  A  =  { y  |  E. z  y A z }
2 nfcv 2279 . . . . 5  |-  F/_ x
y
3 nfrn.1 . . . . 5  |-  F/_ x A
4 nfcv 2279 . . . . 5  |-  F/_ x
z
52, 3, 4nfbr 3969 . . . 4  |-  F/ x  y A z
65nfex 1616 . . 3  |-  F/ x E. z  y A
z
76nfab 2284 . 2  |-  F/_ x { y  |  E. z  y A z }
81, 7nfcxfr 2276 1  |-  F/_ x dom  A
Colors of variables: wff set class
Syntax hints:   E.wex 1468   {cab 2123   F/_wnfc 2266   class class class wbr 3924   dom cdm 4534
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 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-dm 4544
This theorem is referenced by:  nfrn  4779  dmiin  4780  nffn  5214  ellimc3apf  12787
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