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Theorem nffr 4859
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 4844 . 2  |-  ( R  Fr  A  <->  A. a
( ( a  C_  A  /\  a  =/=  (/) )  ->  E. b  e.  a  A. c  e.  a  -.  c R b ) )
2 nfcv 2629 . . . . . 6  |-  F/_ x
a
3 nffr.a . . . . . 6  |-  F/_ x A
42, 3nfss 3502 . . . . 5  |-  F/ x  a  C_  A
5 nfv 1683 . . . . 5  |-  F/ x  a  =/=  (/)
64, 5nfan 1875 . . . 4  |-  F/ x
( a  C_  A  /\  a  =/=  (/) )
7 nfcv 2629 . . . . . . . 8  |-  F/_ x
c
8 nffr.r . . . . . . . 8  |-  F/_ x R
9 nfcv 2629 . . . . . . . 8  |-  F/_ x
b
107, 8, 9nfbr 4497 . . . . . . 7  |-  F/ x  c R b
1110nfn 1849 . . . . . 6  |-  F/ x  -.  c R b
122, 11nfral 2853 . . . . 5  |-  F/ x A. c  e.  a  -.  c R b
132, 12nfrex 2930 . . . 4  |-  F/ x E. b  e.  a  A. c  e.  a  -.  c R b
146, 13nfim 1867 . . 3  |-  F/ x
( ( a  C_  A  /\  a  =/=  (/) )  ->  E. b  e.  a  A. c  e.  a  -.  c R b )
1514nfal 1894 . 2  |-  F/ x A. a ( ( a 
C_  A  /\  a  =/=  (/) )  ->  E. b  e.  a  A. c  e.  a  -.  c R b )
161, 15nfxfr 1625 1  |-  F/ x  R  Fr  A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1377   F/wnf 1599   F/_wnfc 2615    =/= wne 2662   A.wral 2817   E.wrex 2818    C_ wss 3481   (/)c0 3790   class class class wbr 4453    Fr wfr 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-fr 4844
This theorem is referenced by:  nfwe  4861
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