Theorem nfwe 4445

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 4424	. 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 4439	. . 3 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴
5		nfcv 2372	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑎
6		nfcv 2372	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑏
7	5, 2, 6	nfbr 4129	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
8		nfcv 2372	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑐
9	6, 2, 8	nfbr 4129	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑏𝑅𝑐
10	7, 9	nfan 1611	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐)
11	5, 2, 8	nfbr 4129	. . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑐
12	10, 11	nfim 1618	. . . . . 6 ⊢ Ⅎ𝑥((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
13	3, 12	nfralxy 2568	. . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
14	3, 13	nfralxy 2568	. . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
15	3, 14	nfralxy 2568	. . 3 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
16	4, 15	nfan 1611	. 2 ⊢ Ⅎ𝑥(𝑅 Fr 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
17	1, 16	nfxfr 1520	1 ⊢ Ⅎ𝑥 𝑅 We 𝐴

Syntax hints:   → wi 4   ∧ wa 104  Ⅎwnf 1506  Ⅎwnfc 2359  ∀wral 2508   class class class wbr 4082   Fr wfr 4418   We wwe 4420

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-frfor 4421  df-frind 4422  df-wetr 4424

This theorem is referenced by: (None)