Theorem nffrfor 4493

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 4476	. 2 ⊢ ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆) → 𝐴 ⊆ 𝑆))
2		nffrfor.a	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfcv 2392	. . . . . . . 8 ⊢ Ⅎ𝑥𝑣
4		nffrfor.r	. . . . . . . 8 ⊢ Ⅎ𝑥𝑅
5		nfcv 2392	. . . . . . . 8 ⊢ Ⅎ𝑥𝑢
6	3, 4, 5	nfbr 4177	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣𝑅𝑢
7		nffrfor.s	. . . . . . . 8 ⊢ Ⅎ𝑥𝑆
8	7	nfcri 2386	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣 ∈ 𝑆
9	6, 8	nfim 1625	. . . . . 6 ⊢ Ⅎ𝑥(𝑣𝑅𝑢 → 𝑣 ∈ 𝑆)
10	2, 9	nfralxy 2588	. . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆)
11	7	nfcri 2386	. . . . 5 ⊢ Ⅎ𝑥 𝑢 ∈ 𝑆
12	10, 11	nfim 1625	. . . 4 ⊢ Ⅎ𝑥(∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆)
13	2, 12	nfralxy 2588	. . 3 ⊢ Ⅎ𝑥∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆)
14	2, 7	nfss 3241	. . 3 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝑆
15	13, 14	nfim 1625	. 2 ⊢ Ⅎ𝑥(∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆) → 𝐴 ⊆ 𝑆)
16	1, 15	nfxfr 1527	1 ⊢ Ⅎ𝑥 FrFor 𝑅𝐴𝑆

This proof depends on syntax axioms:   → wi 4  Ⅎwnf 1513   ∈ wcel 2209  Ⅎwnfc 2379  ∀wral 2528   ⊆ wss 3220   class class class wbr 4130   FrFor wfrfor 4472

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-frfor 4476

This theorem is used by:  nffr  4494