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Theorem nffrfor 4175
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
nffrfor.r 𝑥𝑅
nffrfor.a 𝑥𝐴
nffrfor.s 𝑥𝑆
Assertion
Ref Expression
nffrfor 𝑥 FrFor 𝑅𝐴𝑆

Proof of Theorem nffrfor
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-frfor 4158 . 2 ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑢𝐴 (∀𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆) → 𝑢𝑆) → 𝐴𝑆))
2 nffrfor.a . . . 4 𝑥𝐴
3 nfcv 2228 . . . . . . . 8 𝑥𝑣
4 nffrfor.r . . . . . . . 8 𝑥𝑅
5 nfcv 2228 . . . . . . . 8 𝑥𝑢
63, 4, 5nfbr 3889 . . . . . . 7 𝑥 𝑣𝑅𝑢
7 nffrfor.s . . . . . . . 8 𝑥𝑆
87nfcri 2222 . . . . . . 7 𝑥 𝑣𝑆
96, 8nfim 1509 . . . . . 6 𝑥(𝑣𝑅𝑢𝑣𝑆)
102, 9nfralxy 2414 . . . . 5 𝑥𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆)
117nfcri 2222 . . . . 5 𝑥 𝑢𝑆
1210, 11nfim 1509 . . . 4 𝑥(∀𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆) → 𝑢𝑆)
132, 12nfralxy 2414 . . 3 𝑥𝑢𝐴 (∀𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆) → 𝑢𝑆)
142, 7nfss 3018 . . 3 𝑥 𝐴𝑆
1513, 14nfim 1509 . 2 𝑥(∀𝑢𝐴 (∀𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆) → 𝑢𝑆) → 𝐴𝑆)
161, 15nfxfr 1408 1 𝑥 FrFor 𝑅𝐴𝑆
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1394  wcel 1438  wnfc 2215  wral 2359  wss 2999   class class class wbr 3845   FrFor wfrfor 4154
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-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-frfor 4158
This theorem is referenced by:  nffr  4176
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