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Theorem nffr 4862
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 4847 . 2  |-  ( R  Fr  A  <->  A. a
( ( a  C_  A  /\  a  =/=  (/) )  ->  E. b  e.  a  A. c  e.  a  -.  c R b ) )
2 nfcv 2619 . . . . . 6  |-  F/_ x
a
3 nffr.a . . . . . 6  |-  F/_ x A
42, 3nfss 3492 . . . . 5  |-  F/ x  a  C_  A
5 nfv 1708 . . . . 5  |-  F/ x  a  =/=  (/)
64, 5nfan 1929 . . . 4  |-  F/ x
( a  C_  A  /\  a  =/=  (/) )
7 nfcv 2619 . . . . . . . 8  |-  F/_ x
c
8 nffr.r . . . . . . . 8  |-  F/_ x R
9 nfcv 2619 . . . . . . . 8  |-  F/_ x
b
107, 8, 9nfbr 4500 . . . . . . 7  |-  F/ x  c R b
1110nfn 1902 . . . . . 6  |-  F/ x  -.  c R b
122, 11nfral 2843 . . . . 5  |-  F/ x A. c  e.  a  -.  c R b
132, 12nfrex 2920 . . . 4  |-  F/ x E. b  e.  a  A. c  e.  a  -.  c R b
146, 13nfim 1921 . . 3  |-  F/ x
( ( a  C_  A  /\  a  =/=  (/) )  ->  E. b  e.  a  A. c  e.  a  -.  c R b )
1514nfal 1948 . 2  |-  F/ x A. a ( ( a 
C_  A  /\  a  =/=  (/) )  ->  E. b  e.  a  A. c  e.  a  -.  c R b )
161, 15nfxfr 1646 1  |-  F/ x  R  Fr  A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1393   F/wnf 1617   F/_wnfc 2605    =/= wne 2652   A.wral 2807   E.wrex 2808    C_ wss 3471   (/)c0 3793   class class class wbr 4456    Fr wfr 4844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-fr 4847
This theorem is referenced by:  nfwe  4864
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