Theorem nfso 5536

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 5530	. 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 5535	. . 3 ⊢ Ⅎ𝑥 𝑅 Po 𝐴
5		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑥𝑎
6		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑥𝑏
7	5, 2, 6	nfbr 5122	. . . . . 6 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
8		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑥 𝑎 = 𝑏
9	6, 2, 5	nfbr 5122	. . . . . 6 ⊢ Ⅎ𝑥 𝑏𝑅𝑎
10	7, 8, 9	nf3or 1913	. . . . 5 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)
11	3, 10	nfralw 3288	. . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)
12	3, 11	nfralw 3288	. . 3 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)
13	4, 12	nfan 1907	. 2 ⊢ Ⅎ𝑥(𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎))
14	1, 13	nfxfr 1861	1 ⊢ Ⅎ𝑥 𝑅 Or 𝐴

Syntax hints:   ∧ wa 397   ∨ w3o 1092  Ⅎwnf 1791  Ⅎwnfc 2888  ∀wral 3055   class class class wbr 5075   Po wpo 5527   Or wor 5528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-po 5529  df-so 5530

This theorem is referenced by:  nfwe  5596  weiunso  36709