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Theorem nfdm 4845
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 4611 . 2  |-  dom  A  =  { y  |  E. z  y A z }
2 nfcv 2306 . . . . 5  |-  F/_ x
y
3 nfrn.1 . . . . 5  |-  F/_ x A
4 nfcv 2306 . . . . 5  |-  F/_ x
z
52, 3, 4nfbr 4025 . . . 4  |-  F/ x  y A z
65nfex 1624 . . 3  |-  F/ x E. z  y A
z
76nfab 2311 . 2  |-  F/_ x { y  |  E. z  y A z }
81, 7nfcxfr 2303 1  |-  F/_ x dom  A
Colors of variables: wff set class
Syntax hints:   E.wex 1479   {cab 2150   F/_wnfc 2293   class class class wbr 3979   dom cdm 4601
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2726  df-un 3118  df-sn 3579  df-pr 3580  df-op 3582  df-br 3980  df-dm 4611
This theorem is referenced by:  nfrn  4846  dmiin  4847  nffn  5281  ellimc3apf  13227
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