Theorem euotd 4353

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

Hypotheses

Ref	Expression
euotd.1	⊢ (𝜑 → 𝐴 ∈ V)
euotd.2	⊢ (𝜑 → 𝐵 ∈ V)
euotd.3	⊢ (𝜑 → 𝐶 ∈ V)
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)

Proof of Theorem euotd

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euotd.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
2		euotd.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
3		euotd.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ V)
4		otexg 4328	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ V)
5	1, 2, 3, 4	syl3anc 1274	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵, 𝐶⟩ ∈ V)
6		euotd.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝜓 ↔ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶)))
7	6	biimpa 296	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶))
8		vex 2806	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
9		vex 2806	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V
10		vex 2806	. . . . . . . . . . . . 13 ⊢ 𝑐 ∈ V
11	8, 9, 10	otth 4340	. . . . . . . . . . . 12 ⊢ (⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩ ↔ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶))
12	7, 11	sylibr 134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩)
13	12	eqeq2d 2243	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
14	13	biimpd 144	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
15	14	impancom 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝑎, 𝑏, 𝑐⟩) → (𝜓 → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
16	15	expimpd 363	. . . . . . 7 ⊢ (𝜑 → ((𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
17	16	exlimdv 1867	. . . . . 6 ⊢ (𝜑 → (∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
18	17	exlimdvv 1946	. . . . 5 ⊢ (𝜑 → (∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
19		tru 1402	. . . . . . . . . . 11 ⊢ ⊤
20	2	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → 𝐵 ∈ V)
21	3	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝐶 ∈ V)
22		simpr 110	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶)) → (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶))
23	22, 11	sylibr 134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶)) → ⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩)
24	23	eqcomd 2237	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶)) → ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩)
25	6	biimpar 297	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶)) → 𝜓)
26	24, 25	jca 306	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶)) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
27		trud 1414	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶)) → ⊤)
28	26, 27	2thd 175	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶)) → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
29	28	3anassrs 1256	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
30	21, 29	sbcied 3069	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → ([𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
31	20, 30	sbcied 3069	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ([𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
32	1, 31	sbcied 3069	. . . . . . . . . . 11 ⊢ (𝜑 → ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
33	19, 32	mpbiri 168	. . . . . . . . . 10 ⊢ (𝜑 → [𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
34	33	spesbcd 3120	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑎[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
35		nfcv 2375	. . . . . . . . . 10 ⊢ Ⅎ𝑏𝐵
36		nfsbc1v 3051	. . . . . . . . . . 11 ⊢ Ⅎ𝑏[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
37	36	nfex 1686	. . . . . . . . . 10 ⊢ Ⅎ𝑏∃𝑎[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
38		sbceq1a 3042	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → ([𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ [𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
39	38	exbidv 1873	. . . . . . . . . 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)))
40	35, 37, 39	spcegf 2890	. . . . . . . . 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)))
41	2, 34, 40	sylc 62	. . . . . . . 8 ⊢ (𝜑 → ∃𝑏∃𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
42		nfcv 2375	. . . . . . . . 9 ⊢ Ⅎ𝑐𝐶
43		nfsbc1v 3051	. . . . . . . . . . 11 ⊢ Ⅎ𝑐[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
44	43	nfex 1686	. . . . . . . . . 10 ⊢ Ⅎ𝑐∃𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
45	44	nfex 1686	. . . . . . . . 9 ⊢ Ⅎ𝑐∃𝑏∃𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
46		sbceq1a 3042	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ [𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
47	46	2exbidv 1916	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (∃𝑏∃𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑏∃𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
48	42, 45, 47	spcegf 2890	. . . . . . . 8 ⊢ (𝐶 ∈ V → (∃𝑏∃𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → ∃𝑐∃𝑏∃𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
49	3, 41, 48	sylc 62	. . . . . . 7 ⊢ (𝜑 → ∃𝑐∃𝑏∃𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
50		excom13 1737	. . . . . . 7 ⊢ (∃𝑐∃𝑏∃𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑎∃𝑏∃𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
51	49, 50	sylib 122	. . . . . 6 ⊢ (𝜑 → ∃𝑎∃𝑏∃𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
52		eqeq1 2238	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ↔ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩))
53	52	anbi1d 465	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → ((𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
54	53	3exbidv 1917	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → (∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑎∃𝑏∃𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
55	51, 54	syl5ibrcom 157	. . . . 5 ⊢ (𝜑 → (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → ∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
56	18, 55	impbid 129	. . . 4 ⊢ (𝜑 → (∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
57	56	alrimiv 1922	. . 3 ⊢ (𝜑 → ∀𝑥(∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
58		eqeq2 2241	. . . . . 6 ⊢ (𝑦 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑥 = 𝑦 ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
59	58	bibi2d 232	. . . . 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⟩)))
60	59	albidv 1872	. . . 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⟩)))
61	60	spcegv 2895	. . 3 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ ∈ V → (∀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)))
62	5, 57, 61	sylc 62	. 2 ⊢ (𝜑 → ∃𝑦∀𝑥(∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦))
63		df-eu 2082	. 2 ⊢ (∃!𝑥∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑦∀𝑥(∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦))
64	62, 63	sylibr 134	1 ⊢ (𝜑 → ∃!𝑥∃𝑎∃𝑏∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005  ∀wal 1396   = wceq 1398  ⊤wtru 1399  ∃wex 1541  ∃!weu 2079   ∈ wcel 2202  Vcvv 2803  [wsbc 3032  ⟨cotp 3677

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-ot 3683

This theorem is referenced by: (None)