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Theorem nffr 4559
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
nffr.r  |-  F/_ x R
nffr.a  |-  F/_ x A
Assertion
Ref Expression
nffr  |-  F/ x  R  Fr  A

Proof of Theorem nffr
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fr 4544 . 2  |-  ( R  Fr  A  <->  A. a
( ( a  C_  A  /\  a  =/=  (/) )  ->  E. b  e.  a  A. c  e.  a  -.  c R b ) )
2 nfcv 2574 . . . . . 6  |-  F/_ x
a
3 nffr.a . . . . . 6  |-  F/_ x A
42, 3nfss 3343 . . . . 5  |-  F/ x  a  C_  A
5 nfv 1630 . . . . 5  |-  F/ x  a  =/=  (/)
64, 5nfan 1847 . . . 4  |-  F/ x
( a  C_  A  /\  a  =/=  (/) )
7 nfcv 2574 . . . . . . . 8  |-  F/_ x
c
8 nffr.r . . . . . . . 8  |-  F/_ x R
9 nfcv 2574 . . . . . . . 8  |-  F/_ x
b
107, 8, 9nfbr 4259 . . . . . . 7  |-  F/ x  c R b
1110nfn 1812 . . . . . 6  |-  F/ x  -.  c R b
122, 11nfral 2761 . . . . 5  |-  F/ x A. c  e.  a  -.  c R b
132, 12nfrex 2763 . . . 4  |-  F/ x E. b  e.  a  A. c  e.  a  -.  c R b
146, 13nfim 1833 . . 3  |-  F/ x
( ( a  C_  A  /\  a  =/=  (/) )  ->  E. b  e.  a  A. c  e.  a  -.  c R b )
1514nfal 1865 . 2  |-  F/ x A. a ( ( a 
C_  A  /\  a  =/=  (/) )  ->  E. b  e.  a  A. c  e.  a  -.  c R b )
161, 15nfxfr 1580 1  |-  F/ x  R  Fr  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360   A.wal 1550   F/wnf 1554   F/_wnfc 2561    =/= wne 2601   A.wral 2707   E.wrex 2708    C_ wss 3322   (/)c0 3630   class class class wbr 4215    Fr wfr 4541
This theorem is referenced by:  nfwe  4561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-fr 4544
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