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Theorem nffrfor 4113
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 4096 . 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 2194 . . . . . . . 8  |-  F/_ x
v
4 nffrfor.r . . . . . . . 8  |-  F/_ x R
5 nfcv 2194 . . . . . . . 8  |-  F/_ x u
63, 4, 5nfbr 3836 . . . . . . 7  |-  F/ x  v R u
7 nffrfor.s . . . . . . . 8  |-  F/_ x S
87nfcri 2188 . . . . . . 7  |-  F/ x  v  e.  S
96, 8nfim 1480 . . . . . 6  |-  F/ x
( v R u  ->  v  e.  S
)
102, 9nfralxy 2377 . . . . 5  |-  F/ x A. v  e.  A  ( v R u  ->  v  e.  S
)
117nfcri 2188 . . . . 5  |-  F/ x  u  e.  S
1210, 11nfim 1480 . . . 4  |-  F/ x
( A. v  e.  A  ( v R u  ->  v  e.  S )  ->  u  e.  S )
132, 12nfralxy 2377 . . 3  |-  F/ x A. u  e.  A  ( A. v  e.  A  ( v R u  ->  v  e.  S
)  ->  u  e.  S )
142, 7nfss 2966 . . 3  |-  F/ x  A  C_  S
1513, 14nfim 1480 . 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 1379 1  |-  F/ xFrFor  R A S
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1365    e. wcel 1409   F/_wnfc 2181   A.wral 2323    C_ wss 2945   class class class wbr 3792  FrFor wfrfor 4092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-frfor 4096
This theorem is referenced by:  nffr  4114
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