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Theorem nfdm 5023
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 4802 . 2  |-  dom  A  =  { y  |  E. z  y A z }
2 nfcv 2502 . . . . 5  |-  F/_ x
y
3 nfrn.1 . . . . 5  |-  F/_ x A
4 nfcv 2502 . . . . 5  |-  F/_ x
z
52, 3, 4nfbr 4169 . . . 4  |-  F/ x  y A z
65nfex 1853 . . 3  |-  F/ x E. z  y A
z
76nfab 2506 . 2  |-  F/_ x { y  |  E. z  y A z }
81, 7nfcxfr 2499 1  |-  F/_ x dom  A
Colors of variables: wff set class
Syntax hints:   E.wex 1546   {cab 2352   F/_wnfc 2489   class class class wbr 4125   dom cdm 4792
This theorem is referenced by:  nfrn  5024  dmiin  5025  nffn  5445  funimass4f  23447  itgsinexplem1  27254  nfdfat  27501  bnj1398  28828  bnj1491  28851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-dm 4802
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