Theorem nfwe 4410

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 4389	. 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 4404	. . 3 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴
5		nfcv 2349	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑎
6		nfcv 2349	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑏
7	5, 2, 6	nfbr 4098	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
8		nfcv 2349	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑐
9	6, 2, 8	nfbr 4098	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑏𝑅𝑐
10	7, 9	nfan 1589	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐)
11	5, 2, 8	nfbr 4098	. . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑐
12	10, 11	nfim 1596	. . . . . 6 ⊢ Ⅎ𝑥((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
13	3, 12	nfralxy 2545	. . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
14	3, 13	nfralxy 2545	. . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
15	3, 14	nfralxy 2545	. . 3 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
16	4, 15	nfan 1589	. 2 ⊢ Ⅎ𝑥(𝑅 Fr 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
17	1, 16	nfxfr 1498	1 ⊢ Ⅎ𝑥 𝑅 We 𝐴

Syntax hints:   → wi 4   ∧ wa 104  Ⅎwnf 1484  Ⅎwnfc 2336  ∀wral 2485   class class class wbr 4051   Fr wfr 4383   We wwe 4385

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-frfor 4386  df-frind 4387  df-wetr 4389

This theorem is referenced by: (None)