Theorem nfso 5509

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 5504	. 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 5508	. . 3 ⊢ Ⅎ𝑥 𝑅 Po 𝐴
5		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑥𝑎
6		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑥𝑏
7	5, 2, 6	nfbr 5121	. . . . . 6 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
8		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑥 𝑎 = 𝑏
9	6, 2, 5	nfbr 5121	. . . . . 6 ⊢ Ⅎ𝑥 𝑏𝑅𝑎
10	7, 8, 9	nf3or 1908	. . . . 5 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)
11	3, 10	nfralw 3151	. . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)
12	3, 11	nfralw 3151	. . 3 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)
13	4, 12	nfan 1902	. 2 ⊢ Ⅎ𝑥(𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎))
14	1, 13	nfxfr 1855	1 ⊢ Ⅎ𝑥 𝑅 Or 𝐴

Syntax hints:   ∧ wa 396   ∨ w3o 1085  Ⅎwnf 1786  Ⅎwnfc 2887  ∀wral 3064   class class class wbr 5074   Po wpo 5501   Or wor 5502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-po 5503  df-so 5504

This theorem is referenced by:  nfwe  5565