Theorem nfpo 5532

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 5526	. 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)))
2		nfpo.a	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcv 2901	. . . . . . . 8 ⊢ Ⅎ𝑥𝑎
4		nfpo.r	. . . . . . . 8 ⊢ Ⅎ𝑥𝑅
5	3, 4, 3	nfbr 5119	. . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑎
6	5	nfn 1864	. . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑎𝑅𝑎
7		nfcv 2901	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑏
8	3, 4, 7	nfbr 5119	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑏
9		nfcv 2901	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑐
10	7, 4, 9	nfbr 5119	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑏𝑅𝑐
11	8, 10	nfan 1906	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐)
12	3, 4, 9	nfbr 5119	. . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑐
13	11, 12	nfim 1903	. . . . . 6 ⊢ Ⅎ𝑥((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
14	6, 13	nfan 1906	. . . . 5 ⊢ Ⅎ𝑥(¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
15	2, 14	nfralw 3286	. . . 4 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
16	2, 15	nfralw 3286	. . 3 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
17	2, 16	nfralw 3286	. 2 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
18	1, 17	nfxfr 1860	1 ⊢ Ⅎ𝑥 𝑅 Po 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  Ⅎwnf 1790  Ⅎwnfc 2886  ∀wral 3053   class class class wbr 5072   Po wpo 5524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-po 5526

This theorem is referenced by:  nfso  5533  weiunpo  36693