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Theorem nfse 5418
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 5403 . 2 (𝑅 Se 𝐴 ↔ ∀𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V)
2 nffr.a . . 3 𝑥𝐴
3 nfcv 2949 . . . . . 6 𝑥𝑎
4 nffr.r . . . . . 6 𝑥𝑅
5 nfcv 2949 . . . . . 6 𝑥𝑏
63, 4, 5nfbr 5009 . . . . 5 𝑥 𝑎𝑅𝑏
76, 2nfrab 3345 . . . 4 𝑥{𝑎𝐴𝑎𝑅𝑏}
87nfel1 2963 . . 3 𝑥{𝑎𝐴𝑎𝑅𝑏} ∈ V
92, 8nfral 3191 . 2 𝑥𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V
101, 9nfxfr 1834 1 𝑥 𝑅 Se 𝐴
Colors of variables: wff setvar class
Syntax hints:  wnf 1765  wcel 2081  wnfc 2933  wral 3105  {crab 3109  Vcvv 3437   class class class wbr 4962   Se wse 5400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-se 5403
This theorem is referenced by:  nfoi  8824
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