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Theorem nfse 4768
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 4753 . 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 2544 . . . . . 6  |-  F/_ x
a
4 nffr.r . . . . . 6  |-  F/_ x R
5 nfcv 2544 . . . . . 6  |-  F/_ x
b
63, 4, 5nfbr 4411 . . . . 5  |-  F/ x  a R b
76, 2nfrab 2964 . . . 4  |-  F/_ x { a  e.  A  |  a R b }
87nfel1 2560 . . 3  |-  F/ x { a  e.  A  |  a R b }  e.  _V
92, 8nfral 2768 . 2  |-  F/ x A. b  e.  A  { a  e.  A  |  a R b }  e.  _V
101, 9nfxfr 1653 1  |-  F/ x  R Se  A
Colors of variables: wff setvar class
Syntax hints:   F/wnf 1624    e. wcel 1826   F/_wnfc 2530   A.wral 2732   {crab 2736   _Vcvv 3034   class class class wbr 4367   Se wse 4750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-se 4753
This theorem is referenced by:  nfoi  7854
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