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Theorem nfse 4368
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 4353 . 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 2419 . . . . . 6  |-  F/_ x
a
4 nffr.r . . . . . 6  |-  F/_ x R
5 nfcv 2419 . . . . . 6  |-  F/_ x
b
63, 4, 5nfbr 4067 . . . . 5  |-  F/ x  a R b
76, 2nfrab 2721 . . . 4  |-  F/_ x { a  e.  A  |  a R b }
87nfel1 2429 . . 3  |-  F/ x { a  e.  A  |  a R b }  e.  _V
92, 8nfral 2596 . 2  |-  F/ x A. b  e.  A  { a  e.  A  |  a R b }  e.  _V
101, 9nfxfr 1557 1  |-  F/ x  R Se  A
Colors of variables: wff set class
Syntax hints:   F/wnf 1531    e. wcel 1684   F/_wnfc 2406   A.wral 2543   {crab 2547   _Vcvv 2788   class class class wbr 4023   Se wse 4350
This theorem is referenced by:  nfoi  7229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-se 4353
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