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Theorem nfse 4335
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 4327 . 2 (𝑅 Se 𝐴 ↔ ∀𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V)
2 nfse.a . . 3 𝑥𝐴
3 nfcv 2317 . . . . . 6 𝑥𝑎
4 nfse.r . . . . . 6 𝑥𝑅
5 nfcv 2317 . . . . . 6 𝑥𝑏
63, 4, 5nfbr 4044 . . . . 5 𝑥 𝑎𝑅𝑏
76, 2nfrabxy 2655 . . . 4 𝑥{𝑎𝐴𝑎𝑅𝑏}
87nfel1 2328 . . 3 𝑥{𝑎𝐴𝑎𝑅𝑏} ∈ V
92, 8nfralxy 2513 . 2 𝑥𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V
101, 9nfxfr 1472 1 𝑥 𝑅 Se 𝐴
Colors of variables: wff set class
Syntax hints:  wnf 1458  wcel 2146  wnfc 2304  wral 2453  {crab 2457  Vcvv 2735   class class class wbr 3998   Se wse 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rab 2462  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-se 4327
This theorem is referenced by: (None)
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