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Theorem nfse 5013
 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 4998 . 2 (𝑅 Se 𝐴 ↔ ∀𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V)
2 nffr.a . . 3 𝑥𝐴
3 nfcv 2751 . . . . . 6 𝑥𝑎
4 nffr.r . . . . . 6 𝑥𝑅
5 nfcv 2751 . . . . . 6 𝑥𝑏
63, 4, 5nfbr 4629 . . . . 5 𝑥 𝑎𝑅𝑏
76, 2nfrab 3100 . . . 4 𝑥{𝑎𝐴𝑎𝑅𝑏}
87nfel1 2765 . . 3 𝑥{𝑎𝐴𝑎𝑅𝑏} ∈ V
92, 8nfral 2929 . 2 𝑥𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V
101, 9nfxfr 1771 1 𝑥 𝑅 Se 𝐴
 Colors of variables: wff setvar class Syntax hints:  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738  ∀wral 2896  {crab 2900  Vcvv 3173   class class class wbr 4583   Se wse 4995 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-se 4998 This theorem is referenced by:  nfoi  8302
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