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Theorem nfse 4168
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
nfse.r 𝑥𝑅
nfse.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 4160 . 2 (𝑅 Se 𝐴 ↔ ∀𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V)
2 nfse.a . . 3 𝑥𝐴
3 nfcv 2228 . . . . . 6 𝑥𝑎
4 nfse.r . . . . . 6 𝑥𝑅
5 nfcv 2228 . . . . . 6 𝑥𝑏
63, 4, 5nfbr 3889 . . . . 5 𝑥 𝑎𝑅𝑏
76, 2nfrabxy 2547 . . . 4 𝑥{𝑎𝐴𝑎𝑅𝑏}
87nfel1 2239 . . 3 𝑥{𝑎𝐴𝑎𝑅𝑏} ∈ V
92, 8nfralxy 2414 . 2 𝑥𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V
101, 9nfxfr 1408 1 𝑥 𝑅 Se 𝐴
Colors of variables: wff set class
Syntax hints:  wnf 1394  wcel 1438  wnfc 2215  wral 2359  {crab 2363  Vcvv 2619   class class class wbr 3845   Se wse 4156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-se 4160
This theorem is referenced by: (None)
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