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Theorem nfse 4498
Description: Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nffr.r  |-  F/_ x R
nffr.a  |-  F/_ x A
Assertion
Ref Expression
nfse  |-  F/ x  R Se  A

Proof of Theorem nfse
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-se 4483 . 2  |-  ( R Se  A  <->  A. b  e.  A  { a  e.  A  |  a R b }  e.  _V )
2 nffr.a . . 3  |-  F/_ x A
3 nfcv 2523 . . . . . 6  |-  F/_ x
a
4 nffr.r . . . . . 6  |-  F/_ x R
5 nfcv 2523 . . . . . 6  |-  F/_ x
b
63, 4, 5nfbr 4197 . . . . 5  |-  F/ x  a R b
76, 2nfrab 2832 . . . 4  |-  F/_ x { a  e.  A  |  a R b }
87nfel1 2533 . . 3  |-  F/ x { a  e.  A  |  a R b }  e.  _V
92, 8nfral 2702 . 2  |-  F/ x A. b  e.  A  { a  e.  A  |  a R b }  e.  _V
101, 9nfxfr 1576 1  |-  F/ x  R Se  A
Colors of variables: wff set class
Syntax hints:   F/wnf 1550    e. wcel 1717   F/_wnfc 2510   A.wral 2649   {crab 2653   _Vcvv 2899   class class class wbr 4153   Se wse 4480
This theorem is referenced by:  nfoi  7416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-se 4483
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