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Theorem nfdm 5114
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 4891 . 2  |-  dom  A  =  { y  |  E. z  y A z }
2 nfcv 2574 . . . . 5  |-  F/_ x
y
3 nfrn.1 . . . . 5  |-  F/_ x A
4 nfcv 2574 . . . . 5  |-  F/_ x
z
52, 3, 4nfbr 4259 . . . 4  |-  F/ x  y A z
65nfex 1866 . . 3  |-  F/ x E. z  y A
z
76nfab 2578 . 2  |-  F/_ x { y  |  E. z  y A z }
81, 7nfcxfr 2571 1  |-  F/_ x dom  A
Colors of variables: wff set class
Syntax hints:   E.wex 1551   {cab 2424   F/_wnfc 2561   class class class wbr 4215   dom cdm 4881
This theorem is referenced by:  nfrn  5115  dmiin  5116  nffn  5544  funimass4f  24049  itgsinexplem1  27738  nfdfat  27984  bnj1398  29477  bnj1491  29500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-dm 4891
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