Theorem nfso 4445

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 4440	. 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 4444	. . 3 ⊢ Ⅎ𝑥 𝑅 Po 𝐴
5		nfcv 2392	. . . . . . . 8 ⊢ Ⅎ𝑥𝑎
6		nfcv 2392	. . . . . . . 8 ⊢ Ⅎ𝑥𝑏
7	5, 2, 6	nfbr 4175	. . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
8		nfcv 2392	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑐
9	5, 2, 8	nfbr 4175	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑐
10	8, 2, 6	nfbr 4175	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑐𝑅𝑏
11	9, 10	nfor 1627	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏)
12	7, 11	nfim 1625	. . . . . 6 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))
13	3, 12	nfralxy 2588	. . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))
14	3, 13	nfralxy 2588	. . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))
15	3, 14	nfralxy 2588	. . 3 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))
16	4, 15	nfan 1618	. 2 ⊢ Ⅎ𝑥(𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏)))
17	1, 16	nfxfr 1527	1 ⊢ Ⅎ𝑥 𝑅 Or 𝐴

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 720  Ⅎwnf 1513  Ⅎwnfc 2379  ∀wral 2528   class class class wbr 4128   Po wpo 4437   Or wor 4438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-po 4439  df-iso 4440

This theorem is referenced by: (None)