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Theorem nffrfor 4347
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 4330 . 2 ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑢𝐴 (∀𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆) → 𝑢𝑆) → 𝐴𝑆))
2 nffrfor.a . . . 4 𝑥𝐴
3 nfcv 2319 . . . . . . . 8 𝑥𝑣
4 nffrfor.r . . . . . . . 8 𝑥𝑅
5 nfcv 2319 . . . . . . . 8 𝑥𝑢
63, 4, 5nfbr 4048 . . . . . . 7 𝑥 𝑣𝑅𝑢
7 nffrfor.s . . . . . . . 8 𝑥𝑆
87nfcri 2313 . . . . . . 7 𝑥 𝑣𝑆
96, 8nfim 1572 . . . . . 6 𝑥(𝑣𝑅𝑢𝑣𝑆)
102, 9nfralxy 2515 . . . . 5 𝑥𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆)
117nfcri 2313 . . . . 5 𝑥 𝑢𝑆
1210, 11nfim 1572 . . . 4 𝑥(∀𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆) → 𝑢𝑆)
132, 12nfralxy 2515 . . 3 𝑥𝑢𝐴 (∀𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆) → 𝑢𝑆)
142, 7nfss 3148 . . 3 𝑥 𝐴𝑆
1513, 14nfim 1572 . 2 𝑥(∀𝑢𝐴 (∀𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆) → 𝑢𝑆) → 𝐴𝑆)
161, 15nfxfr 1474 1 𝑥 FrFor 𝑅𝐴𝑆
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1460  wcel 2148  wnfc 2306  wral 2455  wss 3129   class class class wbr 4002   FrFor wfrfor 4326
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-frfor 4330
This theorem is referenced by:  nffr  4348
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