Theorem nfpo 5546

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 5540	. 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)))
2		nfpo.a	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥𝑎
4		nfpo.r	. . . . . . . 8 ⊢ Ⅎ𝑥𝑅
5	3, 4, 3	nfbr 5147	. . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑎
6	5	nfn 1859	. . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑎𝑅𝑎
7		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑏
8	3, 4, 7	nfbr 5147	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
9		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑐
10	7, 4, 9	nfbr 5147	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑏𝑅𝑐
11	8, 10	nfan 1901	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐)
12	3, 4, 9	nfbr 5147	. . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑐
13	11, 12	nfim 1898	. . . . . 6 ⊢ Ⅎ𝑥((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
14	6, 13	nfan 1901	. . . . 5 ⊢ Ⅎ𝑥(¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
15	2, 14	nfralw 3285	. . . 4 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
16	2, 15	nfralw 3285	. . 3 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
17	2, 16	nfralw 3285	. 2 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
18	1, 17	nfxfr 1855	1 ⊢ Ⅎ𝑥 𝑅 Po 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  Ⅎwnf 1785  Ⅎwnfc 2884  ∀wral 3052   class class class wbr 5100   Po wpo 5538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-po 5540

This theorem is referenced by:  nfso  5547  weiunpo  36681