Theorem nffr 4104

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 variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-frind 4087	. 2 |- ( R Fr A <-> A. sFrFor R A s )
2		nffr.r	. . . 4 |- F/_ x R
3		nffr.a	. . . 4 |- F/_ x A
4		nfcv 2219	. . . 4 |- F/_ x s
5	2, 3, 4	nffrfor 4103	. . 3 |- F/ xFrFor R A s
6	5	nfal 1508	. 2 |- F/ x A. sFrFor R A s
7	1, 6	nfxfr 1403	1 |- F/ x R Fr A

Syntax hints:   A.wal 1282   F/wnf 1389   F/_wnfc 2206  FrFor wfrfor 4082   Fr wfr 4083

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-frfor 4086  df-frind 4087

This theorem is referenced by:  nfwe  4110