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Theorem nfse 5118
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 𝑥𝑅
nffr.a 𝑥𝐴
Assertion
Ref Expression
nfse 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 5103 . 2 (𝑅 Se 𝐴 ↔ ∀𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V)
2 nffr.a . . 3 𝑥𝐴
3 nfcv 2793 . . . . . 6 𝑥𝑎
4 nffr.r . . . . . 6 𝑥𝑅
5 nfcv 2793 . . . . . 6 𝑥𝑏
63, 4, 5nfbr 4732 . . . . 5 𝑥 𝑎𝑅𝑏
76, 2nfrab 3153 . . . 4 𝑥{𝑎𝐴𝑎𝑅𝑏}
87nfel1 2808 . . 3 𝑥{𝑎𝐴𝑎𝑅𝑏} ∈ V
92, 8nfral 2974 . 2 𝑥𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V
101, 9nfxfr 1819 1 𝑥 𝑅 Se 𝐴
Colors of variables: wff setvar class
Syntax hints:  wnf 1748  wcel 2030  wnfc 2780  wral 2941  {crab 2945  Vcvv 3231   class class class wbr 4685   Se wse 5100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-se 5103
This theorem is referenced by:  nfoi  8460
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