Theorem nfwe 4206

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 4185	. 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 4200	. . 3 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴
5		nfcv 2235	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑎
6		nfcv 2235	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑏
7	5, 2, 6	nfbr 3911	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
8		nfcv 2235	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑐
9	6, 2, 8	nfbr 3911	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑏𝑅𝑐
10	7, 9	nfan 1509	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐)
11	5, 2, 8	nfbr 3911	. . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑐
12	10, 11	nfim 1516	. . . . . 6 ⊢ Ⅎ𝑥((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
13	3, 12	nfralxy 2425	. . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
14	3, 13	nfralxy 2425	. . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
15	3, 14	nfralxy 2425	. . 3 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
16	4, 15	nfan 1509	. 2 ⊢ Ⅎ𝑥(𝑅 Fr 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
17	1, 16	nfxfr 1415	1 ⊢ Ⅎ𝑥 𝑅 We 𝐴

Syntax hints:   → wi 4   ∧ wa 103  Ⅎwnf 1401  Ⅎwnfc 2222  ∀wral 2370   class class class wbr 3867   Fr wfr 4179   We wwe 4181

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-frfor 4182  df-frind 4183  df-wetr 4185

This theorem is referenced by: (None)