Theorem nffrfor 4451

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 4434	. 2 ⊢ ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆) → 𝐴 ⊆ 𝑆))
2		nffrfor.a	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfcv 2375	. . . . . . . 8 ⊢ Ⅎ𝑥𝑣
4		nffrfor.r	. . . . . . . 8 ⊢ Ⅎ𝑥𝑅
5		nfcv 2375	. . . . . . . 8 ⊢ Ⅎ𝑥𝑢
6	3, 4, 5	nfbr 4140	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣𝑅𝑢
7		nffrfor.s	. . . . . . . 8 ⊢ Ⅎ𝑥𝑆
8	7	nfcri 2369	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣 ∈ 𝑆
9	6, 8	nfim 1621	. . . . . 6 ⊢ Ⅎ𝑥(𝑣𝑅𝑢 → 𝑣 ∈ 𝑆)
10	2, 9	nfralxy 2571	. . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆)
11	7	nfcri 2369	. . . . 5 ⊢ Ⅎ𝑥 𝑢 ∈ 𝑆
12	10, 11	nfim 1621	. . . 4 ⊢ Ⅎ𝑥(∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆)
13	2, 12	nfralxy 2571	. . 3 ⊢ Ⅎ𝑥∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆)
14	2, 7	nfss 3221	. . 3 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝑆
15	13, 14	nfim 1621	. 2 ⊢ Ⅎ𝑥(∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆) → 𝐴 ⊆ 𝑆)
16	1, 15	nfxfr 1523	1 ⊢ Ⅎ𝑥 FrFor 𝑅𝐴𝑆

Syntax hints:   → wi 4  Ⅎwnf 1509   ∈ wcel 2202  Ⅎwnfc 2362  ∀wral 2511   ⊆ wss 3201   class class class wbr 4093   FrFor wfrfor 4430

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-frfor 4434

This theorem is referenced by:  nffr  4452