Theorem nfwe 4386

Description: Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
nfwe.r	⊢ Ⅎ𝑥𝑅
nfwe.a	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfwe	⊢ Ⅎ𝑥 𝑅 We 𝐴

Proof of Theorem nfwe

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

Step	Hyp	Ref	Expression
1		df-wetr 4365	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)))
2		nfwe.r	. . . 4 ⊢ Ⅎ𝑥𝑅
3		nfwe.a	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nffr 4380	. . 3 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴
5		nfcv 2336	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑎
6		nfcv 2336	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑏
7	5, 2, 6	nfbr 4075	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
8		nfcv 2336	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑐
9	6, 2, 8	nfbr 4075	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑏𝑅𝑐
10	7, 9	nfan 1576	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐)
11	5, 2, 8	nfbr 4075	. . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑐
12	10, 11	nfim 1583	. . . . . 6 ⊢ Ⅎ𝑥((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
13	3, 12	nfralxy 2532	. . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
14	3, 13	nfralxy 2532	. . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
15	3, 14	nfralxy 2532	. . 3 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
16	4, 15	nfan 1576	. 2 ⊢ Ⅎ𝑥(𝑅 Fr 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
17	1, 16	nfxfr 1485	1 ⊢ Ⅎ𝑥 𝑅 We 𝐴

Syntax hints:   → wi 4   ∧ wa 104  Ⅎwnf 1471  Ⅎwnfc 2323  ∀wral 2472   class class class wbr 4029   Fr wfr 4359   We wwe 4361

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-frfor 4362  df-frind 4363  df-wetr 4365

This theorem is referenced by: (None)