Theorem nfso 5474

Description: Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Hypotheses

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

Assertion

Ref	Expression
nfso	⊢ Ⅎ𝑥 𝑅 Or 𝐴

Proof of Theorem nfso

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

Step	Hyp	Ref	Expression
1		df-so 5469	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)))
2		nfpo.r	. . . 4 ⊢ Ⅎ𝑥𝑅
3		nfpo.a	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfpo 5473	. . 3 ⊢ Ⅎ𝑥 𝑅 Po 𝐴
5		nfcv 2977	. . . . . . 7 ⊢ Ⅎ𝑥𝑎
6		nfcv 2977	. . . . . . 7 ⊢ Ⅎ𝑥𝑏
7	5, 2, 6	nfbr 5105	. . . . . 6 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
8		nfv 1911	. . . . . 6 ⊢ Ⅎ𝑥 𝑎 = 𝑏
9	6, 2, 5	nfbr 5105	. . . . . 6 ⊢ Ⅎ𝑥 𝑏𝑅𝑎
10	7, 8, 9	nf3or 1902	. . . . 5 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)
11	3, 10	nfralw 3225	. . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)
12	3, 11	nfralw 3225	. . 3 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)
13	4, 12	nfan 1896	. 2 ⊢ Ⅎ𝑥(𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎))
14	1, 13	nfxfr 1849	1 ⊢ Ⅎ𝑥 𝑅 Or 𝐴

Syntax hints:   ∧ wa 398   ∨ w3o 1082  Ⅎwnf 1780  Ⅎwnfc 2961  ∀wral 3138   class class class wbr 5058   Po wpo 5466   Or wor 5467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-po 5468  df-so 5469

This theorem is referenced by:  nfwe  5525