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Theorem nffrfor 4270
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  |-  F/_ x R
nffrfor.a  |-  F/_ x A
nffrfor.s  |-  F/_ x S
Assertion
Ref Expression
nffrfor  |-  F/ xFrFor  R A S

Proof of Theorem nffrfor
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-frfor 4253 . 2  |-  (FrFor  R A S  <->  ( A. u  e.  A  ( A. v  e.  A  (
v R u  -> 
v  e.  S )  ->  u  e.  S
)  ->  A  C_  S
) )
2 nffrfor.a . . . 4  |-  F/_ x A
3 nfcv 2281 . . . . . . . 8  |-  F/_ x
v
4 nffrfor.r . . . . . . . 8  |-  F/_ x R
5 nfcv 2281 . . . . . . . 8  |-  F/_ x u
63, 4, 5nfbr 3974 . . . . . . 7  |-  F/ x  v R u
7 nffrfor.s . . . . . . . 8  |-  F/_ x S
87nfcri 2275 . . . . . . 7  |-  F/ x  v  e.  S
96, 8nfim 1551 . . . . . 6  |-  F/ x
( v R u  ->  v  e.  S
)
102, 9nfralxy 2471 . . . . 5  |-  F/ x A. v  e.  A  ( v R u  ->  v  e.  S
)
117nfcri 2275 . . . . 5  |-  F/ x  u  e.  S
1210, 11nfim 1551 . . . 4  |-  F/ x
( A. v  e.  A  ( v R u  ->  v  e.  S )  ->  u  e.  S )
132, 12nfralxy 2471 . . 3  |-  F/ x A. u  e.  A  ( A. v  e.  A  ( v R u  ->  v  e.  S
)  ->  u  e.  S )
142, 7nfss 3090 . . 3  |-  F/ x  A  C_  S
1513, 14nfim 1551 . 2  |-  F/ x
( A. u  e.  A  ( A. v  e.  A  ( v R u  ->  v  e.  S )  ->  u  e.  S )  ->  A  C_  S )
161, 15nfxfr 1450 1  |-  F/ xFrFor  R A S
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1436    e. wcel 1480   F/_wnfc 2268   A.wral 2416    C_ wss 3071   class class class wbr 3929  FrFor wfrfor 4249
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-frfor 4253
This theorem is referenced by:  nffr  4271
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