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Theorem nffr 4520
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 4505 . 2  |-  ( R  Fr  A  <->  A. a
( ( a  C_  A  /\  a  =/=  (/) )  ->  E. b  e.  a  A. c  e.  a  -.  c R b ) )
2 nfcv 2544 . . . . . 6  |-  F/_ x
a
3 nffr.a . . . . . 6  |-  F/_ x A
42, 3nfss 3305 . . . . 5  |-  F/ x  a  C_  A
5 nfv 1626 . . . . 5  |-  F/ x  a  =/=  (/)
64, 5nfan 1842 . . . 4  |-  F/ x
( a  C_  A  /\  a  =/=  (/) )
7 nfcv 2544 . . . . . . . 8  |-  F/_ x
c
8 nffr.r . . . . . . . 8  |-  F/_ x R
9 nfcv 2544 . . . . . . . 8  |-  F/_ x
b
107, 8, 9nfbr 4220 . . . . . . 7  |-  F/ x  c R b
1110nfn 1807 . . . . . 6  |-  F/ x  -.  c R b
122, 11nfral 2723 . . . . 5  |-  F/ x A. c  e.  a  -.  c R b
132, 12nfrex 2725 . . . 4  |-  F/ x E. b  e.  a  A. c  e.  a  -.  c R b
146, 13nfim 1828 . . 3  |-  F/ x
( ( a  C_  A  /\  a  =/=  (/) )  ->  E. b  e.  a  A. c  e.  a  -.  c R b )
1514nfal 1860 . 2  |-  F/ x A. a ( ( a 
C_  A  /\  a  =/=  (/) )  ->  E. b  e.  a  A. c  e.  a  -.  c R b )
161, 15nfxfr 1576 1  |-  F/ x  R  Fr  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1546   F/wnf 1550   F/_wnfc 2531    =/= wne 2571   A.wral 2670   E.wrex 2671    C_ wss 3284   (/)c0 3592   class class class wbr 4176    Fr wfr 4502
This theorem is referenced by:  nfwe  4522
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-fr 4505
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