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Theorem nfse 4106
 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 4098 . 2 (𝑅 Se 𝐴 ↔ ∀𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V)
2 nfse.a . . 3 𝑥𝐴
3 nfcv 2194 . . . . . 6 𝑥𝑎
4 nfse.r . . . . . 6 𝑥𝑅
5 nfcv 2194 . . . . . 6 𝑥𝑏
63, 4, 5nfbr 3836 . . . . 5 𝑥 𝑎𝑅𝑏
76, 2nfrabxy 2507 . . . 4 𝑥{𝑎𝐴𝑎𝑅𝑏}
87nfel1 2204 . . 3 𝑥{𝑎𝐴𝑎𝑅𝑏} ∈ V
92, 8nfralxy 2377 . 2 𝑥𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V
101, 9nfxfr 1379 1 𝑥 𝑅 Se 𝐴
 Colors of variables: wff set class Syntax hints:  Ⅎwnf 1365   ∈ wcel 1409  Ⅎwnfc 2181  ∀wral 2323  {crab 2327  Vcvv 2574   class class class wbr 3792   Se wse 4094 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rab 2332  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-se 4098 This theorem is referenced by: (None)
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