Theorem nffr 4446

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 4429	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
2		nffr.r	. . . 4 ⊢ Ⅎ𝑥𝑅
3		nffr.a	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfcv 2374	. . . 4 ⊢ Ⅎ𝑥𝑠
5	2, 3, 4	nffrfor 4445	. . 3 ⊢ Ⅎ𝑥 FrFor 𝑅𝐴𝑠
6	5	nfal 1624	. 2 ⊢ Ⅎ𝑥∀𝑠 FrFor 𝑅𝐴𝑠
7	1, 6	nfxfr 1522	1 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴

Syntax hints:  ∀wal 1395  Ⅎwnf 1508  Ⅎwnfc 2361   FrFor wfrfor 4424   Fr wfr 4425

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-frfor 4428  df-frind 4429

This theorem is referenced by:  nfwe  4452