Theorem nffr 5563

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 variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fr 5544	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏))
2		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑥𝑎
3		nffr.a	. . . . . 6 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfss 3913	. . . . 5 ⊢ Ⅎ𝑥 𝑎 ⊆ 𝐴
5		nfv 1917	. . . . 5 ⊢ Ⅎ𝑥 𝑎 ≠ ∅
6	4, 5	nfan 1902	. . . 4 ⊢ Ⅎ𝑥(𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)
7		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥𝑐
8		nffr.r	. . . . . . . 8 ⊢ Ⅎ𝑥𝑅
9		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥𝑏
10	7, 8, 9	nfbr 5121	. . . . . . 7 ⊢ Ⅎ𝑥 𝑐𝑅𝑏
11	10	nfn 1860	. . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑐𝑅𝑏
12	2, 11	nfralw 3151	. . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏
13	2, 12	nfrex 3242	. . . 4 ⊢ Ⅎ𝑥∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏
14	6, 13	nfim 1899	. . 3 ⊢ Ⅎ𝑥((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏)
15	14	nfal 2317	. 2 ⊢ Ⅎ𝑥∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏)
16	1, 15	nfxfr 1855	1 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1537  Ⅎwnf 1786  Ⅎwnfc 2887   ≠ wne 2943  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887  ∅c0 4256   class class class wbr 5074   Fr wfr 5541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-fr 5544

This theorem is referenced by:  nfwe  5565