Theorem nffr 4380

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 4363	. 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 2336	. . . 4 |- F/_ x s
5	2, 3, 4	nffrfor 4379	. . 3 |- F/ xFrFor R A s
6	5	nfal 1587	. 2 |- F/ x A. sFrFor R A s
7	1, 6	nfxfr 1485	1 |- F/ x R Fr A

Syntax hints:   A.wal 1362   F/wnf 1471   F/_wnfc 2323  FrFor wfrfor 4358   Fr wfr 4359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-frfor 4362  df-frind 4363

This theorem is referenced by:  nfwe  4386