Theorem nffr 4266

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 4249	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
2		nffr.r	. . . 4 ⊢ Ⅎ𝑥𝑅
3		nffr.a	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfcv 2279	. . . 4 ⊢ Ⅎ𝑥𝑠
5	2, 3, 4	nffrfor 4265	. . 3 ⊢ Ⅎ𝑥 FrFor 𝑅𝐴𝑠
6	5	nfal 1555	. 2 ⊢ Ⅎ𝑥∀𝑠 FrFor 𝑅𝐴𝑠
7	1, 6	nfxfr 1450	1 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴

Syntax hints:  ∀wal 1329  Ⅎwnf 1436  Ⅎwnfc 2266   FrFor wfrfor 4244   Fr wfr 4245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-frfor 4248  df-frind 4249

This theorem is referenced by:  nfwe  4272