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Theorem nfse 4078
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 4070 . 2 (𝑅 Se 𝐴 ↔ ∀𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V)
2 nfse.a . . 3 𝑥𝐴
3 nfcv 2178 . . . . . 6 𝑥𝑎
4 nfse.r . . . . . 6 𝑥𝑅
5 nfcv 2178 . . . . . 6 𝑥𝑏
63, 4, 5nfbr 3808 . . . . 5 𝑥 𝑎𝑅𝑏
76, 2nfrabxy 2490 . . . 4 𝑥{𝑎𝐴𝑎𝑅𝑏}
87nfel1 2188 . . 3 𝑥{𝑎𝐴𝑎𝑅𝑏} ∈ V
92, 8nfralxy 2360 . 2 𝑥𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V
101, 9nfxfr 1363 1 𝑥 𝑅 Se 𝐴
Colors of variables: wff set class
Syntax hints:  wnf 1349  wcel 1393  wnfc 2165  wral 2306  {crab 2310  Vcvv 2557   class class class wbr 3764   Se wse 4066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-se 4070
This theorem is referenced by: (None)
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