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Theorem nffr 4367
Description: Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nffr.r 𝑥𝑅
nffr.a 𝑥𝐴
Assertion
Ref Expression
nffr 𝑥 𝑅 Fr 𝐴

Proof of Theorem nffr
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 df-frind 4350 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
2 nffr.r . . . 4 𝑥𝑅
3 nffr.a . . . 4 𝑥𝐴
4 nfcv 2332 . . . 4 𝑥𝑠
52, 3, 4nffrfor 4366 . . 3 𝑥 FrFor 𝑅𝐴𝑠
65nfal 1587 . 2 𝑥𝑠 FrFor 𝑅𝐴𝑠
71, 6nfxfr 1485 1 𝑥 𝑅 Fr 𝐴
Colors of variables: wff set class
Syntax hints:  wal 1362  wnf 1471  wnfc 2319   FrFor wfrfor 4345   Fr wfr 4346
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-frfor 4349  df-frind 4350
This theorem is referenced by:  nfwe  4373
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