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Theorem nffrfor 4238
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 4221 . 2 ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑢𝐴 (∀𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆) → 𝑢𝑆) → 𝐴𝑆))
2 nffrfor.a . . . 4 𝑥𝐴
3 nfcv 2256 . . . . . . . 8 𝑥𝑣
4 nffrfor.r . . . . . . . 8 𝑥𝑅
5 nfcv 2256 . . . . . . . 8 𝑥𝑢
63, 4, 5nfbr 3942 . . . . . . 7 𝑥 𝑣𝑅𝑢
7 nffrfor.s . . . . . . . 8 𝑥𝑆
87nfcri 2250 . . . . . . 7 𝑥 𝑣𝑆
96, 8nfim 1534 . . . . . 6 𝑥(𝑣𝑅𝑢𝑣𝑆)
102, 9nfralxy 2446 . . . . 5 𝑥𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆)
117nfcri 2250 . . . . 5 𝑥 𝑢𝑆
1210, 11nfim 1534 . . . 4 𝑥(∀𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆) → 𝑢𝑆)
132, 12nfralxy 2446 . . 3 𝑥𝑢𝐴 (∀𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆) → 𝑢𝑆)
142, 7nfss 3058 . . 3 𝑥 𝐴𝑆
1513, 14nfim 1534 . 2 𝑥(∀𝑢𝐴 (∀𝑣𝐴 (𝑣𝑅𝑢𝑣𝑆) → 𝑢𝑆) → 𝐴𝑆)
161, 15nfxfr 1433 1 𝑥 FrFor 𝑅𝐴𝑆
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1419  wcel 1463  wnfc 2243  wral 2391  wss 3039   class class class wbr 3897   FrFor wfrfor 4217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-frfor 4221
This theorem is referenced by:  nffr  4239
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