Theorem nfso 4348

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 4343	. 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 4347	. . 3 ⊢ Ⅎ𝑥 𝑅 Po 𝐴
5		nfcv 2347	. . . . . . . 8 ⊢ Ⅎ𝑥𝑎
6		nfcv 2347	. . . . . . . 8 ⊢ Ⅎ𝑥𝑏
7	5, 2, 6	nfbr 4089	. . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
8		nfcv 2347	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑐
9	5, 2, 8	nfbr 4089	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑐
10	8, 2, 6	nfbr 4089	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑐𝑅𝑏
11	9, 10	nfor 1596	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏)
12	7, 11	nfim 1594	. . . . . 6 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))
13	3, 12	nfralxy 2543	. . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))
14	3, 13	nfralxy 2543	. . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))
15	3, 14	nfralxy 2543	. . 3 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))
16	4, 15	nfan 1587	. 2 ⊢ Ⅎ𝑥(𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏)))
17	1, 16	nfxfr 1496	1 ⊢ Ⅎ𝑥 𝑅 Or 𝐴

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709  Ⅎwnf 1482  Ⅎwnfc 2334  ∀wral 2483   class class class wbr 4043   Po wpo 4340   Or wor 4341

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-po 4342  df-iso 4343

This theorem is referenced by: (None)