Theorem nffr 4384

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 4367	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
2		nffr.r	. . . 4 ⊢ Ⅎ𝑥𝑅
3		nffr.a	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfcv 2339	. . . 4 ⊢ Ⅎ𝑥𝑠
5	2, 3, 4	nffrfor 4383	. . 3 ⊢ Ⅎ𝑥 FrFor 𝑅𝐴𝑠
6	5	nfal 1590	. 2 ⊢ Ⅎ𝑥∀𝑠 FrFor 𝑅𝐴𝑠
7	1, 6	nfxfr 1488	1 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴

Syntax hints:  ∀wal 1362  Ⅎwnf 1474  Ⅎwnfc 2326   FrFor wfrfor 4362   Fr wfr 4363

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-frfor 4366  df-frind 4367

This theorem is referenced by:  nfwe  4390