Theorem nffr 5088

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 5073	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏))
2		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑥𝑎
3		nffr.a	. . . . . 6 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfss 3596	. . . . 5 ⊢ Ⅎ𝑥 𝑎 ⊆ 𝐴
5		nfv 1843	. . . . 5 ⊢ Ⅎ𝑥 𝑎 ≠ ∅
6	4, 5	nfan 1828	. . . 4 ⊢ Ⅎ𝑥(𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅)
7		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑥𝑐
8		nffr.r	. . . . . . . 8 ⊢ Ⅎ𝑥𝑅
9		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑥𝑏
10	7, 8, 9	nfbr 4699	. . . . . . 7 ⊢ Ⅎ𝑥 𝑐𝑅𝑏
11	10	nfn 1784	. . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑐𝑅𝑏
12	2, 11	nfral 2945	. . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏
13	2, 12	nfrex 3007	. . . 4 ⊢ Ⅎ𝑥∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏
14	6, 13	nfim 1825	. . 3 ⊢ Ⅎ𝑥((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏)
15	14	nfal 2153	. 2 ⊢ Ⅎ𝑥∀𝑎((𝑎 ⊆ 𝐴 ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑅𝑏)
16	1, 15	nfxfr 1779	1 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384  ∀wal 1481  Ⅎwnf 1708  Ⅎwnfc 2751   ≠ wne 2794  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574  ∅c0 3915   class class class wbr 4653   Fr wfr 5070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-fr 5073

This theorem is referenced by:  nfwe  5090