Theorem nffr 5582

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 5563	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏))
2		nfcv 2905	. . . . . 6 ⊢ Ⅎ𝑥𝑎
3		nffr.a	. . . . . 6 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfss 3923	. . . . 5 ⊢ Ⅎ𝑥 𝑎 ⊆ 𝐴
5		nfv 1916	. . . . 5 ⊢ Ⅎ𝑥 𝑎 ≠ ∅
6	4, 5	nfan 1901	. . . 4 ⊢ Ⅎ𝑥(𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)
7		nfcv 2905	. . . . . . . 8 ⊢ Ⅎ𝑥𝑐
8		nffr.r	. . . . . . . 8 ⊢ Ⅎ𝑥𝑅
9		nfcv 2905	. . . . . . . 8 ⊢ Ⅎ𝑥𝑏
10	7, 8, 9	nfbr 5134	. . . . . . 7 ⊢ Ⅎ𝑥 𝑐𝑅𝑏
11	10	nfn 1859	. . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑐𝑅𝑏
12	2, 11	nfralw 3291	. . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏
13	2, 12	nfrexw 3293	. . . 4 ⊢ Ⅎ𝑥∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏
14	6, 13	nfim 1898	. . 3 ⊢ Ⅎ𝑥((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏)
15	14	nfal 2316	. 2 ⊢ Ⅎ𝑥∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏)
16	1, 15	nfxfr 1854	1 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1538  Ⅎwnf 1784  Ⅎwnfc 2885   ≠ wne 2941  ∀wral 3062  ∃wrex 3071   ⊆ wss 3897  ∅c0 4267   class class class wbr 5087   Fr wfr 5560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-sn 4572  df-pr 4574  df-op 4578  df-br 5088  df-fr 5563

This theorem is referenced by:  nfwe  5584