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Theorem nfse 5608
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 5589 . 2 (𝑅 Se 𝐴 ↔ ∀𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V)
2 nffr.a . . 3 𝑥𝐴
3 nfcv 2907 . . . . . 6 𝑥𝑎
4 nffr.r . . . . . 6 𝑥𝑅
5 nfcv 2907 . . . . . 6 𝑥𝑏
63, 4, 5nfbr 5152 . . . . 5 𝑥 𝑎𝑅𝑏
76, 2nfrabw 3440 . . . 4 𝑥{𝑎𝐴𝑎𝑅𝑏}
87nfel1 2923 . . 3 𝑥{𝑎𝐴𝑎𝑅𝑏} ∈ V
92, 8nfralw 3294 . 2 𝑥𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V
101, 9nfxfr 1855 1 𝑥 𝑅 Se 𝐴
Colors of variables: wff setvar class
Syntax hints:  wnf 1785  wcel 2106  wnfc 2887  wral 3064  {crab 3407  Vcvv 3445   class class class wbr 5105   Se wse 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-se 5589
This theorem is referenced by:  nfoi  9449
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