Theorem nffr 4367

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	⊢ Ⅎ𝑥𝑅
nffr.a	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nffr	⊢ Ⅎ𝑥 𝑅 Fr 𝐴

Proof of Theorem nffr

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-frind 4350	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
2		nffr.r	. . . 4 ⊢ Ⅎ𝑥𝑅
3		nffr.a	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfcv 2332	. . . 4 ⊢ Ⅎ𝑥𝑠
5	2, 3, 4	nffrfor 4366	. . 3 ⊢ Ⅎ𝑥 FrFor 𝑅𝐴𝑠
6	5	nfal 1587	. 2 ⊢ Ⅎ𝑥∀𝑠 FrFor 𝑅𝐴𝑠
7	1, 6	nfxfr 1485	1 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴

Syntax hints:  ∀wal 1362  Ⅎwnf 1471  Ⅎwnfc 2319   FrFor wfrfor 4345   Fr wfr 4346

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 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-frfor 4349  df-frind 4350

This theorem is referenced by:  nfwe  4373