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Theorem nfdm 5250
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 5015 . 2  |-  dom  A  =  { y  |  E. z  y A z }
2 nfcv 2629 . . . . 5  |-  F/_ x
y
3 nfrn.1 . . . . 5  |-  F/_ x A
4 nfcv 2629 . . . . 5  |-  F/_ x
z
52, 3, 4nfbr 4497 . . . 4  |-  F/ x  y A z
65nfex 1895 . . 3  |-  F/ x E. z  y A
z
76nfab 2633 . 2  |-  F/_ x { y  |  E. z  y A z }
81, 7nfcxfr 2627 1  |-  F/_ x dom  A
Colors of variables: wff setvar class
Syntax hints:   E.wex 1596   {cab 2452   F/_wnfc 2615   class class class wbr 4453   dom cdm 5005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-dm 5015
This theorem is referenced by:  nfrn  5251  dmiin  5252  nffn  5683  funimass4f  27297  itgsinexplem1  31594  fourierdlem16  31746  fourierdlem21  31751  fourierdlem22  31752  fourierdlem68  31798  fourierdlem80  31810  fourierdlem103  31833  fourierdlem104  31834  nfdfat  32005  bnj1398  33570  bnj1491  33593
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