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Theorem nfdm 5078
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 4855 . 2  |-  dom  A  =  { y  |  E. z  y A z }
2 nfcv 2548 . . . . 5  |-  F/_ x
y
3 nfrn.1 . . . . 5  |-  F/_ x A
4 nfcv 2548 . . . . 5  |-  F/_ x
z
52, 3, 4nfbr 4224 . . . 4  |-  F/ x  y A z
65nfex 1861 . . 3  |-  F/ x E. z  y A
z
76nfab 2552 . 2  |-  F/_ x { y  |  E. z  y A z }
81, 7nfcxfr 2545 1  |-  F/_ x dom  A
Colors of variables: wff set class
Syntax hints:   E.wex 1547   {cab 2398   F/_wnfc 2535   class class class wbr 4180   dom cdm 4845
This theorem is referenced by:  nfrn  5079  dmiin  5080  nffn  5508  funimass4f  24005  itgsinexplem1  27623  nfdfat  27869  bnj1398  29121  bnj1491  29144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-dm 4855
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