Theorem nfso 4057

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 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iso 4052	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))))
2		nfpo.r	. . . 4 ⊢ Ⅎ𝑥𝑅
3		nfpo.a	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfpo 4056	. . 3 ⊢ Ⅎ𝑥 𝑅 Po 𝐴
5		nfcv 2219	. . . . . . . 8 ⊢ Ⅎ𝑥𝑎
6		nfcv 2219	. . . . . . . 8 ⊢ Ⅎ𝑥𝑏
7	5, 2, 6	nfbr 3829	. . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
8		nfcv 2219	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑐
9	5, 2, 8	nfbr 3829	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑐
10	8, 2, 6	nfbr 3829	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑐𝑅𝑏
11	9, 10	nfor 1506	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏)
12	7, 11	nfim 1504	. . . . . 6 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))
13	3, 12	nfralxy 2402	. . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))
14	3, 13	nfralxy 2402	. . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))
15	3, 14	nfralxy 2402	. . 3 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))
16	4, 15	nfan 1497	. 2 ⊢ Ⅎ𝑥(𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏)))
17	1, 16	nfxfr 1403	1 ⊢ Ⅎ𝑥 𝑅 Or 𝐴

Syntax hints:   → wi 4   ∧ wa 102   ∨ wo 661  Ⅎwnf 1389  Ⅎwnfc 2206  ∀wral 2348   class class class wbr 3785   Po wpo 4049   Or wor 4050

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-po 4051  df-iso 4052

This theorem is referenced by: (None)