Theorem poclOLD 5529

Description: Obsolete version of pocl 5528 as of 3-Oct-2024. (Contributed by NM, 27-Mar-1997.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
poclOLD	⊢ (𝑅 Po 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))))

Proof of Theorem poclOLD

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . . 7 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
2	1, 1	breq12d 5100	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑥 ↔ 𝐵𝑅𝐵))
3	2	notbid 317	. . . . 5 ⊢ (𝑥 = 𝐵 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝐵𝑅𝐵))
4		breq1 5090	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝐵𝑅𝑦))
5	4	anbi1d 630	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧)))
6		breq1 5090	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑧 ↔ 𝐵𝑅𝑧))
7	5, 6	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝐵 → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧)))
8	3, 7	anbi12d 631	. . . 4 ⊢ (𝑥 = 𝐵 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧))))
9	8	imbi2d 340	. . 3 ⊢ (𝑥 = 𝐵 → ((𝑅 Po 𝐴 → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) ↔ (𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧)))))
10		breq2 5091	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝐵𝑅𝑦 ↔ 𝐵𝑅𝐶))
11		breq1 5090	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦𝑅𝑧 ↔ 𝐶𝑅𝑧))
12	10, 11	anbi12d 631	. . . . . 6 ⊢ (𝑦 = 𝐶 → ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧)))
13	12	imbi1d 341	. . . . 5 ⊢ (𝑦 = 𝐶 → (((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧) ↔ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧)))
14	13	anbi2d 629	. . . 4 ⊢ (𝑦 = 𝐶 → ((¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧)) ↔ (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧))))
15	14	imbi2d 340	. . 3 ⊢ (𝑦 = 𝐶 → ((𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧))) ↔ (𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧)))))
16		breq2 5091	. . . . . . 7 ⊢ (𝑧 = 𝐷 → (𝐶𝑅𝑧 ↔ 𝐶𝑅𝐷))
17	16	anbi2d 629	. . . . . 6 ⊢ (𝑧 = 𝐷 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) ↔ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷)))
18		breq2 5091	. . . . . 6 ⊢ (𝑧 = 𝐷 → (𝐵𝑅𝑧 ↔ 𝐵𝑅𝐷))
19	17, 18	imbi12d 344	. . . . 5 ⊢ (𝑧 = 𝐷 → (((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧) ↔ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)))
20	19	anbi2d 629	. . . 4 ⊢ (𝑧 = 𝐷 → ((¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧)) ↔ (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))))
21	20	imbi2d 340	. . 3 ⊢ (𝑧 = 𝐷 → ((𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧))) ↔ (𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)))))
22		df-po 5521	. . . . . . 7 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
23		r3al 3190	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
24	22, 23	sylbb 218	. . . . . 6 ⊢ (𝑅 Po 𝐴 → ∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
25	24	19.21bbi 2182	. . . . 5 ⊢ (𝑅 Po 𝐴 → ∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
26	25	19.21bi 2181	. . . 4 ⊢ (𝑅 Po 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
27	26	com12 32	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑅 Po 𝐴 → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
28	9, 15, 21, 27	vtocl3ga 3526	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))))
29	28	com12 32	1 ⊢ (𝑅 Po 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2105  ∀wral 3062   class class class wbr 5087   Po wpo 5519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rab 3405  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-sn 4572  df-pr 4574  df-op 4578  df-br 5088  df-po 5521

This theorem is referenced by: (None)