Theorem nfso 4965

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 4960	. 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 4964	. . 3 ⊢ Ⅎ𝑥 𝑅 Po 𝐴
5		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑥𝑎
6		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑥𝑏
7	5, 2, 6	nfbr 4629	. . . . . 6 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
8		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑥 𝑎 = 𝑏
9	6, 2, 5	nfbr 4629	. . . . . 6 ⊢ Ⅎ𝑥 𝑏𝑅𝑎
10	7, 8, 9	nf3or 1823	. . . . 5 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)
11	3, 10	nfral 2929	. . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)
12	3, 11	nfral 2929	. . 3 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)
13	4, 12	nfan 1816	. 2 ⊢ Ⅎ𝑥(𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎))
14	1, 13	nfxfr 1771	1 ⊢ Ⅎ𝑥 𝑅 Or 𝐴

Syntax hints:   ∧ wa 383   ∨ w3o 1030  Ⅎwnf 1699  Ⅎwnfc 2738  ∀wral 2896   class class class wbr 4583   Po wpo 4957   Or wor 4958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-po 4959  df-so 4960

This theorem is referenced by:  nfwe  5014