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Theorem nfse 4826
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 4811 . 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 2585 . . . . . 6  |-  F/_ x
a
4 nffr.r . . . . . 6  |-  F/_ x R
5 nfcv 2585 . . . . . 6  |-  F/_ x
b
63, 4, 5nfbr 4466 . . . . 5  |-  F/ x  a R b
76, 2nfrab 3011 . . . 4  |-  F/_ x { a  e.  A  |  a R b }
87nfel1 2601 . . 3  |-  F/ x { a  e.  A  |  a R b }  e.  _V
92, 8nfral 2812 . 2  |-  F/ x A. b  e.  A  { a  e.  A  |  a R b }  e.  _V
101, 9nfxfr 1693 1  |-  F/ x  R Se  A
Colors of variables: wff setvar class
Syntax hints:   F/wnf 1664    e. wcel 1869   F/_wnfc 2571   A.wral 2776   {crab 2780   _Vcvv 3082   class class class wbr 4421   Se wse 4808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ral 2781  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-br 4422  df-se 4811
This theorem is referenced by:  nfoi  8033
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