Theorem nfso 4302

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 4297	. 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 4301	. . 3 ⊢ Ⅎ𝑥 𝑅 Po 𝐴
5		nfcv 2319	. . . . . . . 8 ⊢ Ⅎ𝑥𝑎
6		nfcv 2319	. . . . . . . 8 ⊢ Ⅎ𝑥𝑏
7	5, 2, 6	nfbr 4049	. . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
8		nfcv 2319	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑐
9	5, 2, 8	nfbr 4049	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑐
10	8, 2, 6	nfbr 4049	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑐𝑅𝑏
11	9, 10	nfor 1574	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏)
12	7, 11	nfim 1572	. . . . . 6 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))
13	3, 12	nfralxy 2515	. . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))
14	3, 13	nfralxy 2515	. . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))
15	3, 14	nfralxy 2515	. . 3 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏))
16	4, 15	nfan 1565	. 2 ⊢ Ⅎ𝑥(𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐 ∨ 𝑐𝑅𝑏)))
17	1, 16	nfxfr 1474	1 ⊢ Ⅎ𝑥 𝑅 Or 𝐴

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 708  Ⅎwnf 1460  Ⅎwnfc 2306  ∀wral 2455   class class class wbr 4003   Po wpo 4294   Or wor 4295

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-po 4296  df-iso 4297

This theorem is referenced by: (None)