Theorem nffr 4279

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 4262	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
2		nffr.r	. . . 4 ⊢ Ⅎ𝑥𝑅
3		nffr.a	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfcv 2282	. . . 4 ⊢ Ⅎ𝑥𝑠
5	2, 3, 4	nffrfor 4278	. . 3 ⊢ Ⅎ𝑥 FrFor 𝑅𝐴𝑠
6	5	nfal 1556	. 2 ⊢ Ⅎ𝑥∀𝑠 FrFor 𝑅𝐴𝑠
7	1, 6	nfxfr 1451	1 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴

Syntax hints:  ∀wal 1330  Ⅎwnf 1437  Ⅎwnfc 2269   FrFor wfrfor 4257   Fr wfr 4258

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-frfor 4261  df-frind 4262

This theorem is referenced by:  nfwe  4285