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Theorem nffrfor 4085
 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 4068 . 2 ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑢𝐴 (∀𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆) → 𝑢𝑆) → 𝐴𝑆))
2 nffrfor.a . . . 4 𝑥𝐴
3 nfcv 2178 . . . . . . . 8 𝑥𝑣
4 nffrfor.r . . . . . . . 8 𝑥𝑅
5 nfcv 2178 . . . . . . . 8 𝑥𝑢
63, 4, 5nfbr 3808 . . . . . . 7 𝑥 𝑣𝑅𝑢
7 nffrfor.s . . . . . . . 8 𝑥𝑆
87nfcri 2172 . . . . . . 7 𝑥 𝑣𝑆
96, 8nfim 1464 . . . . . 6 𝑥(𝑣𝑅𝑢𝑣𝑆)
102, 9nfralxy 2360 . . . . 5 𝑥𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆)
117nfcri 2172 . . . . 5 𝑥 𝑢𝑆
1210, 11nfim 1464 . . . 4 𝑥(∀𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆) → 𝑢𝑆)
132, 12nfralxy 2360 . . 3 𝑥𝑢𝐴 (∀𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆) → 𝑢𝑆)
142, 7nfss 2938 . . 3 𝑥 𝐴𝑆
1513, 14nfim 1464 . 2 𝑥(∀𝑢𝐴 (∀𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆) → 𝑢𝑆) → 𝐴𝑆)
161, 15nfxfr 1363 1 𝑥 FrFor 𝑅𝐴𝑆
 Colors of variables: wff set class Syntax hints:   → wi 4  Ⅎwnf 1349   ∈ wcel 1393  Ⅎwnfc 2165  ∀wral 2306   ⊆ wss 2917   class class class wbr 3764   FrFor wfrfor 4064 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-frfor 4068 This theorem is referenced by:  nffr  4086
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