Theorem nffr 4187

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 4170	. 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 2229	. . . 4 |- F/_ x s
5	2, 3, 4	nffrfor 4186	. . 3 |- F/ xFrFor R A s
6	5	nfal 1514	. 2 |- F/ x A. sFrFor R A s
7	1, 6	nfxfr 1409	1 |- F/ x R Fr A

Syntax hints:   A.wal 1288   F/wnf 1395   F/_wnfc 2216  FrFor wfrfor 4165   Fr wfr 4166

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-sn 3458  df-pr 3459  df-op 3461  df-br 3854  df-frfor 4169  df-frind 4170

This theorem is referenced by:  nfwe  4193