Theorem nffrfor 4379

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
nffrfor.r	⊢ Ⅎ𝑥𝑅
nffrfor.a	⊢ Ⅎ𝑥𝐴
nffrfor.s	⊢ Ⅎ𝑥𝑆

Assertion

Ref	Expression
nffrfor	⊢ Ⅎ𝑥 FrFor 𝑅𝐴𝑆

Proof of Theorem nffrfor

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-frfor 4362	. 2 ⊢ ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆) → 𝐴 ⊆ 𝑆))
2		nffrfor.a	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfcv 2336	. . . . . . . 8 ⊢ Ⅎ𝑥𝑣
4		nffrfor.r	. . . . . . . 8 ⊢ Ⅎ𝑥𝑅
5		nfcv 2336	. . . . . . . 8 ⊢ Ⅎ𝑥𝑢
6	3, 4, 5	nfbr 4075	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣𝑅𝑢
7		nffrfor.s	. . . . . . . 8 ⊢ Ⅎ𝑥𝑆
8	7	nfcri 2330	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣 ∈ 𝑆
9	6, 8	nfim 1583	. . . . . 6 ⊢ Ⅎ𝑥(𝑣𝑅𝑢 → 𝑣 ∈ 𝑆)
10	2, 9	nfralxy 2532	. . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆)
11	7	nfcri 2330	. . . . 5 ⊢ Ⅎ𝑥 𝑢 ∈ 𝑆
12	10, 11	nfim 1583	. . . 4 ⊢ Ⅎ𝑥(∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆)
13	2, 12	nfralxy 2532	. . 3 ⊢ Ⅎ𝑥∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆)
14	2, 7	nfss 3172	. . 3 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝑆
15	13, 14	nfim 1583	. 2 ⊢ Ⅎ𝑥(∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆) → 𝐴 ⊆ 𝑆)
16	1, 15	nfxfr 1485	1 ⊢ Ⅎ𝑥 FrFor 𝑅𝐴𝑆

Syntax hints:   → wi 4  Ⅎwnf 1471   ∈ wcel 2164  Ⅎwnfc 2323  ∀wral 2472   ⊆ wss 3153   class class class wbr 4029   FrFor wfrfor 4358

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 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-frfor 4362

This theorem is referenced by:  nffr  4380