Theorem nfpo 4356

Description: Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Hypotheses

Ref	Expression
nfpo.r	⊢ Ⅎ𝑥𝑅
nfpo.a	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfpo	⊢ Ⅎ𝑥 𝑅 Po 𝐴

Proof of Theorem nfpo

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-po 4351	. 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)))
2		nfpo.a	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcv 2349	. . . . . . . 8 ⊢ Ⅎ𝑥𝑎
4		nfpo.r	. . . . . . . 8 ⊢ Ⅎ𝑥𝑅
5	3, 4, 3	nfbr 4098	. . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑎
6	5	nfn 1682	. . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑎𝑅𝑎
7		nfcv 2349	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑏
8	3, 4, 7	nfbr 4098	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
9		nfcv 2349	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑐
10	7, 4, 9	nfbr 4098	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑏𝑅𝑐
11	8, 10	nfan 1589	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐)
12	3, 4, 9	nfbr 4098	. . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑐
13	11, 12	nfim 1596	. . . . . 6 ⊢ Ⅎ𝑥((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
14	6, 13	nfan 1589	. . . . 5 ⊢ Ⅎ𝑥(¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
15	2, 14	nfralxy 2545	. . . 4 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
16	2, 15	nfralxy 2545	. . 3 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
17	2, 16	nfralxy 2545	. 2 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
18	1, 17	nfxfr 1498	1 ⊢ Ⅎ𝑥 𝑅 Po 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  Ⅎwnf 1484  Ⅎwnfc 2336  ∀wral 2485   class class class wbr 4051   Po wpo 4349

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-po 4351

This theorem is referenced by:  nfso  4357