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Theorem nfdm 4637
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 4411 . 2  |-  dom  A  =  { y  |  E. z  y A z }
2 nfcv 2223 . . . . 5  |-  F/_ x
y
3 nfrn.1 . . . . 5  |-  F/_ x A
4 nfcv 2223 . . . . 5  |-  F/_ x
z
52, 3, 4nfbr 3855 . . . 4  |-  F/ x  y A z
65nfex 1569 . . 3  |-  F/ x E. z  y A
z
76nfab 2227 . 2  |-  F/_ x { y  |  E. z  y A z }
81, 7nfcxfr 2220 1  |-  F/_ x dom  A
Colors of variables: wff set class
Syntax hints:   E.wex 1422   {cab 2069   F/_wnfc 2210   class class class wbr 3811   dom cdm 4401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-dm 4411
This theorem is referenced by:  nfrn  4638  dmiin  4639  nffn  5063
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