Theorem euotd 5497

Description: Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.)

Hypotheses

Ref	Expression
euotd.1	⊢ (𝜑 → 𝐴 ∈ 𝑈)
euotd.2	⊢ (𝜑 → 𝐵 ∈ 𝑉)
euotd.3	⊢ (𝜑 → 𝐶 ∈ 𝑊)
euotd.4	⊢ (𝜑 → (𝜓 ↔ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶)))

Assertion

Ref	Expression
euotd	⊢ (𝜑 → ∃!𝑥∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))

Distinct variable groups:   𝑎,𝑏,𝑐,𝑥,𝐴   𝐵,𝑎,𝑏,𝑐,𝑥   𝐶,𝑎,𝑏,𝑐,𝑥   𝜑,𝑎,𝑏,𝑐,𝑥

Allowed substitution hints:   𝜓(𝑥,𝑎,𝑏,𝑐)   𝑈(𝑥,𝑎,𝑏,𝑐)   𝑉(𝑥,𝑎,𝑏,𝑐)   𝑊(𝑥,𝑎,𝑏,𝑐)

Proof of Theorem euotd

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euotd.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝜓 ↔ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶)))
2	1	biimpa 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶))
3		vex 3467	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
4		vex 3467	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V
5		vex 3467	. . . . . . . . . . . . 13 ⊢ 𝑐 ∈ V
6	3, 4, 5	otth 5467	. . . . . . . . . . . 12 ⊢ (⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩ ↔ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶))
7	2, 6	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩)
8	7	eqeq2d 2780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
9	8	biimpd 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
10	9	impancom 456	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝑎, 𝑏, 𝑐⟩) → (𝜓 → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
11	10	expimpd 458	. . . . . . 7 ⊢ (𝜑 → ((𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
12	11	exlimdv 1960	. . . . . 6 ⊢ (𝜑 → (∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
13	12	exlimdvv 1961	. . . . 5 ⊢ (𝜑 → (∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
14		euotd.3	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
15		euotd.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
16		tru 1571	. . . . . . . . . . 11 ⊢ ⊤
17		euotd.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
18	15	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → 𝐵 ∈ 𝑉)
19	14	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝐶 ∈ 𝑊)
20	6	bilanri 511	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶)) → ⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩)
21	20	eqcomd 2775	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶)) → ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩)
22	1	biimpar 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶)) → 𝜓)
23	21, 22	jca 520	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶)) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
24		trud 1577	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶)) → ⊤)
25	23, 24	2thd 268	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶)) → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
26	25	3anassrs 1379	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
27	19, 26	sbcied 3796	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → ([𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
28	18, 27	sbcied 3796	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ([𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
29	17, 28	sbcied 3796	. . . . . . . . . . 11 ⊢ (𝜑 → ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
30	16, 29	mpbiri 261	. . . . . . . . . 10 ⊢ (𝜑 → [𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
31	30	spesbcd 3845	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑎[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
32		nfcv 2931	. . . . . . . . . 10 ⊢ Ⅎ𝑏𝐵
33		nfsbc1v 3773	. . . . . . . . . . 11 ⊢ Ⅎ𝑏[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
34	33	nfex 2363	. . . . . . . . . 10 ⊢ Ⅎ𝑏∃𝑎[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
35		sbceq1a 3764	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → ([𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ [𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
36	35	exbidv 1948	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (∃𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑎[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
37	32, 34, 36	spcegf 3560	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑉 → (∃𝑎[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → ∃𝑏∃𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
38	15, 31, 37	sylc 66	. . . . . . . 8 ⊢ (𝜑 → ∃𝑏∃𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
39		nfcv 2931	. . . . . . . . 9 ⊢ Ⅎ𝑐𝐶
40		nfsbc1v 3773	. . . . . . . . . . 11 ⊢ Ⅎ𝑐[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
41	40	nfex 2363	. . . . . . . . . 10 ⊢ Ⅎ𝑐∃𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
42	41	nfex 2363	. . . . . . . . 9 ⊢ Ⅎ𝑐∃𝑏∃𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
43		sbceq1a 3764	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ [𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
44	43	2exbidv 1951	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (∃𝑏∃𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑏∃𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
45	39, 42, 44	spcegf 3560	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑊 → (∃𝑏∃𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → ∃𝑐∃𝑏∃𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
46	14, 38, 45	sylc 66	. . . . . . 7 ⊢ (𝜑 → ∃𝑐∃𝑏∃𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
47		excom13 2205	. . . . . . 7 ⊢ (∃𝑐∃𝑏∃𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑎∃𝑏∃𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
48	46, 47	sylib 221	. . . . . 6 ⊢ (𝜑 → ∃𝑎∃𝑏∃𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
49		eqeq1 2773	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ↔ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩))
50	49	anbi1d 642	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → ((𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
51	50	3exbidv 1952	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → (∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑎∃𝑏∃𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
52	48, 51	syl5ibrcom 250	. . . . 5 ⊢ (𝜑 → (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → ∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
53	13, 52	impbid 215	. . . 4 ⊢ (𝜑 → (∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
54	53	alrimiv 1954	. . 3 ⊢ (𝜑 → ∀𝑥(∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
55		otex 5448	. . . 4 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ ∈ V
56		eqeq2 2781	. . . . . 6 ⊢ (𝑦 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑥 = 𝑦 ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
57	56	bibi2d 345	. . . . 5 ⊢ (𝑦 = ⟨𝐴, 𝐵, 𝐶⟩ → ((∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦) ↔ (∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩)))
58	57	albidv 1947	. . . 4 ⊢ (𝑦 = ⟨𝐴, 𝐵, 𝐶⟩ → (∀𝑥(∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦) ↔ ∀𝑥(∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩)))
59	55, 58	spcev 3574	. . 3 ⊢ (∀𝑥(∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩) → ∃𝑦∀𝑥(∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦))
60	54, 59	syl 18	. 2 ⊢ (𝜑 → ∃𝑦∀𝑥(∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦))
61		eu6 2608	. 2 ⊢ (∃!𝑥∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑦∀𝑥(∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦))
62	60, 61	sylibr 237	1 ⊢ (𝜑 → ∃!𝑥∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101  ∀wal 1565   = wceq 1567  ⊤wtru 1568  ∃wex 1806   ∈ wcel 2149  ∃!weu 2602  [wsbc 3753  ⟨cotp 4602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-ot 4603

This theorem is referenced by:  oeeu  8588