Theorem oprabid 5853

Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Although this theorem would be useful with a distinct variable condition between 𝑥, 𝑦, and 𝑧, we use ax-bndl 1489 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.)

Assertion

Ref	Expression
oprabid	⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)

Proof of Theorem oprabid

Dummy variables 𝑎 𝑟 𝑠 𝑡 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2715	. . . 4 ⊢ 𝑥 ∈ V
2		vex 2715	. . . 4 ⊢ 𝑦 ∈ V
3	1, 2	opex 4189	. . 3 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
4		vex 2715	. . 3 ⊢ 𝑧 ∈ V
5		opexg 4188	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ V ∧ 𝑧 ∈ V) → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ V)
6	3, 4, 5	mp2an 423	. 2 ⊢ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ V
7	3, 4	eqvinop 4203	. . . . 5 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑎∃𝑡(𝑤 = ⟨𝑎, 𝑡⟩ ∧ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
8	7	biimpi 119	. . . 4 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ∃𝑎∃𝑡(𝑤 = ⟨𝑎, 𝑡⟩ ∧ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
9		eqeq1 2164	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
10		vex 2715	. . . . . . . . 9 ⊢ 𝑎 ∈ V
11		vex 2715	. . . . . . . . 9 ⊢ 𝑡 ∈ V
12	10, 11	opth1 4196	. . . . . . . 8 ⊢ (⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑎 = ⟨𝑥, 𝑦⟩)
13	9, 12	syl6bi 162	. . . . . . 7 ⊢ (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑎 = ⟨𝑥, 𝑦⟩))
14	1, 2	eqvinop 4203	. . . . . . . . 9 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑟∃𝑠(𝑎 = ⟨𝑟, 𝑠⟩ ∧ ⟨𝑟, 𝑠⟩ = ⟨𝑥, 𝑦⟩))
15		opeq1 3741	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨𝑟, 𝑠⟩ → ⟨𝑎, 𝑡⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩)
16	15	eqeq2d 2169	. . . . . . . . . . . 12 ⊢ (𝑎 = ⟨𝑟, 𝑠⟩ → (𝑤 = ⟨𝑎, 𝑡⟩ ↔ 𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩))
17	1, 2, 4	otth2 4201	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ↔ (𝑥 = 𝑟 ∧ 𝑦 = 𝑠 ∧ 𝑧 = 𝑡))
18		df-3an 965	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑟 ∧ 𝑦 = 𝑠 ∧ 𝑧 = 𝑡) ↔ ((𝑥 = 𝑟 ∧ 𝑦 = 𝑠) ∧ 𝑧 = 𝑡))
19	17, 18	bitri 183	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ↔ ((𝑥 = 𝑟 ∧ 𝑦 = 𝑠) ∧ 𝑧 = 𝑡))
20	19	anbi1i 454	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) ↔ (((𝑥 = 𝑟 ∧ 𝑦 = 𝑠) ∧ 𝑧 = 𝑡) ∧ 𝜑))
21		anass 399	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 = 𝑟 ∧ 𝑦 = 𝑠) ∧ 𝑧 = 𝑡) ∧ 𝜑) ↔ ((𝑥 = 𝑟 ∧ 𝑦 = 𝑠) ∧ (𝑧 = 𝑡 ∧ 𝜑)))
22		anass 399	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 = 𝑟 ∧ 𝑦 = 𝑠) ∧ (𝑧 = 𝑡 ∧ 𝜑)) ↔ (𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))))
23	20, 21, 22	3bitri 205	. . . . . . . . . . . . . . . 16 ⊢ ((⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) ↔ (𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))))
24	23	3exbii 1587	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥∃𝑦∃𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))))
25		oprabidlem 5852	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥∃𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))) → ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))))
26	25	eximi 1580	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑦∃𝑥∃𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))) → ∃𝑦∃𝑥(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))))
27		excom 1644	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑥∃𝑦∃𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))) ↔ ∃𝑦∃𝑥∃𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))))
28		excom 1644	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑥∃𝑦(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))) ↔ ∃𝑦∃𝑥(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))))
29	26, 27, 28	3imtr4i 200	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥∃𝑦∃𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))) → ∃𝑥∃𝑦(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))))
30		oprabidlem 5852	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥∃𝑦(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))) → ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))))
31		oprabidlem 5852	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑦∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑)) → ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑)))
32	31	anim2i 340	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑟 ∧ ∃𝑦∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))) → (𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))))
33	32	eximi 1580	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))) → ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))))
34	29, 30, 33	3syl 17	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥∃𝑦∃𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))) → ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))))
35	24, 34	sylbi 120	. . . . . . . . . . . . . 14 ⊢ (∃𝑥∃𝑦∃𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) → ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))))
36		euequ1 2101	. . . . . . . . . . . . . . . . . . 19 ⊢ ∃!𝑥 𝑥 = 𝑟
37		eupick 2085	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∃!𝑥 𝑥 = 𝑟 ∧ ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑)))) → (𝑥 = 𝑟 → ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))))
38	36, 37	mpan 421	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))) → (𝑥 = 𝑟 → ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))))
39		euequ1 2101	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∃!𝑦 𝑦 = 𝑠
40		eupick 2085	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∃!𝑦 𝑦 = 𝑠 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))) → (𝑦 = 𝑠 → ∃𝑧(𝑧 = 𝑡 ∧ 𝜑)))
41	39, 40	mpan 421	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑)) → (𝑦 = 𝑠 → ∃𝑧(𝑧 = 𝑡 ∧ 𝜑)))
42		euequ1 2101	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∃!𝑧 𝑧 = 𝑡
43		eupick 2085	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∃!𝑧 𝑧 = 𝑡 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑)) → (𝑧 = 𝑡 → 𝜑))
44	42, 43	mpan 421	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑧(𝑧 = 𝑡 ∧ 𝜑) → (𝑧 = 𝑡 → 𝜑))
45	41, 44	syl6 33	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑)) → (𝑦 = 𝑠 → (𝑧 = 𝑡 → 𝜑)))
46	38, 45	syl6 33	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))) → (𝑥 = 𝑟 → (𝑦 = 𝑠 → (𝑧 = 𝑡 → 𝜑))))
47	46	3impd 1203	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))) → ((𝑥 = 𝑟 ∧ 𝑦 = 𝑠 ∧ 𝑧 = 𝑡) → 𝜑))
48	17, 47	syl5bi 151	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))) → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → 𝜑))
49	48	com12 30	. . . . . . . . . . . . . 14 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))) → 𝜑))
50	35, 49	syl5 32	. . . . . . . . . . . . 13 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (∃𝑥∃𝑦∃𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) → 𝜑))
51		eqeq1 2164	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
52		eqcom 2159	. . . . . . . . . . . . . . 15 ⊢ (⟨⟨𝑟, 𝑠⟩, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩)
53	51, 52	bitrdi 195	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩))
54	53	anbi1d 461	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑)))
55	54	3exbidv 1849	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑)))
56	55	imbi1d 230	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → ((∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑) ↔ (∃𝑥∃𝑦∃𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) → 𝜑)))
57	53, 56	imbi12d 233	. . . . . . . . . . . . 13 ⊢ (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑)) ↔ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (∃𝑥∃𝑦∃𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) → 𝜑))))
58	50, 57	mpbiri 167	. . . . . . . . . . . 12 ⊢ (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑)))
59	16, 58	syl6bi 162	. . . . . . . . . . 11 ⊢ (𝑎 = ⟨𝑟, 𝑠⟩ → (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))))
60	59	adantr 274	. . . . . . . . . 10 ⊢ ((𝑎 = ⟨𝑟, 𝑠⟩ ∧ ⟨𝑟, 𝑠⟩ = ⟨𝑥, 𝑦⟩) → (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))))
61	60	exlimivv 1876	. . . . . . . . 9 ⊢ (∃𝑟∃𝑠(𝑎 = ⟨𝑟, 𝑠⟩ ∧ ⟨𝑟, 𝑠⟩ = ⟨𝑥, 𝑦⟩) → (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))))
62	14, 61	sylbi 120	. . . . . . . 8 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))))
63	62	com3l 81	. . . . . . 7 ⊢ (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑎 = ⟨𝑥, 𝑦⟩ → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))))
64	13, 63	mpdd 41	. . . . . 6 ⊢ (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑)))
65	64	adantr 274	. . . . 5 ⊢ ((𝑤 = ⟨𝑎, 𝑡⟩ ∧ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑)))
66	65	exlimivv 1876	. . . 4 ⊢ (∃𝑎∃𝑡(𝑤 = ⟨𝑎, 𝑡⟩ ∧ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑)))
67	8, 66	mpcom 36	. . 3 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))
68		19.8a 1570	. . . . 5 ⊢ ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
69		19.8a 1570	. . . . 5 ⊢ (∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
70		19.8a 1570	. . . . 5 ⊢ (∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
71	68, 69, 70	3syl 17	. . . 4 ⊢ ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
72	71	ex 114	. . 3 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑 → ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
73	67, 72	impbid 128	. 2 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ 𝜑))
74		df-oprab 5828	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
75	6, 73, 74	elab2 2860	1 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1335  ∃wex 1472  ∃!weu 2006   ∈ wcel 2128  Vcvv 2712  ⟨cop 3563  {coprab 5825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-setind 4496

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-oprab 5828

This theorem is referenced by:  ssoprab2b  5878  ovid  5937  ovidig  5938  tposoprab  6227  xpcomco  6771