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Theorem nfdm 5096
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 4864 . 2  |-  dom  A  =  { y  |  E. z  y A z }
2 nfcv 2591 . . . . 5  |-  F/_ x
y
3 nfrn.1 . . . . 5  |-  F/_ x A
4 nfcv 2591 . . . . 5  |-  F/_ x
z
52, 3, 4nfbr 4470 . . . 4  |-  F/ x  y A z
65nfex 2006 . . 3  |-  F/ x E. z  y A
z
76nfab 2595 . 2  |-  F/_ x { y  |  E. z  y A z }
81, 7nfcxfr 2589 1  |-  F/_ x dom  A
Colors of variables: wff setvar class
Syntax hints:   E.wex 1659   {cab 2414   F/_wnfc 2577   class class class wbr 4426   dom cdm 4854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-dm 4864
This theorem is referenced by:  nfrn  5097  dmiin  5098  nffn  5690  funimass4f  28074  bnj1398  29631  bnj1491  29654  itgsinexplem1  37402  fourierdlem16  37557  fourierdlem21  37562  fourierdlem22  37563  fourierdlem68  37609  fourierdlem80  37621  fourierdlem103  37644  fourierdlem104  37645  nfdfat  38034
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