Theorem oprabid 5989

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 1533 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 2776	. . . 4 ⊢ 𝑥 ∈ V
2		vex 2776	. . . 4 ⊢ 𝑦 ∈ V
3	1, 2	opex 4281	. . 3 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
4		vex 2776	. . 3 ⊢ 𝑧 ∈ V
5		opexg 4280	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ V ∧ 𝑧 ∈ V) → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ V)
6	3, 4, 5	mp2an 426	. 2 ⊢ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ V
7	3, 4	eqvinop 4295	. . . . 5 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑎∃𝑡(𝑤 = ⟨𝑎, 𝑡⟩ ∧ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
8	7	biimpi 120	. . . 4 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ∃𝑎∃𝑡(𝑤 = ⟨𝑎, 𝑡⟩ ∧ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
9		eqeq1 2213	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
10		vex 2776	. . . . . . . . 9 ⊢ 𝑎 ∈ V
11		vex 2776	. . . . . . . . 9 ⊢ 𝑡 ∈ V
12	10, 11	opth1 4288	. . . . . . . 8 ⊢ (⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑎 = ⟨𝑥, 𝑦⟩)
13	9, 12	biimtrdi 163	. . . . . . 7 ⊢ (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑎 = ⟨𝑥, 𝑦⟩))
14	1, 2	eqvinop 4295	. . . . . . . . 9 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑟∃𝑠(𝑎 = ⟨𝑟, 𝑠⟩ ∧ ⟨𝑟, 𝑠⟩ = ⟨𝑥, 𝑦⟩))
15		opeq1 3825	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨𝑟, 𝑠⟩ → ⟨𝑎, 𝑡⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩)
16	15	eqeq2d 2218	. . . . . . . . . . . 12 ⊢ (𝑎 = ⟨𝑟, 𝑠⟩ → (𝑤 = ⟨𝑎, 𝑡⟩ ↔ 𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩))
17	1, 2, 4	otth2 4293	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ↔ (𝑥 = 𝑟 ∧ 𝑦 = 𝑠 ∧ 𝑧 = 𝑡))
18		df-3an 983	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑟 ∧ 𝑦 = 𝑠 ∧ 𝑧 = 𝑡) ↔ ((𝑥 = 𝑟 ∧ 𝑦 = 𝑠) ∧ 𝑧 = 𝑡))
19	17, 18	bitri 184	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ↔ ((𝑥 = 𝑟 ∧ 𝑦 = 𝑠) ∧ 𝑧 = 𝑡))
20	19	anbi1i 458	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) ↔ (((𝑥 = 𝑟 ∧ 𝑦 = 𝑠) ∧ 𝑧 = 𝑡) ∧ 𝜑))
21		anass 401	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 = 𝑟 ∧ 𝑦 = 𝑠) ∧ 𝑧 = 𝑡) ∧ 𝜑) ↔ ((𝑥 = 𝑟 ∧ 𝑦 = 𝑠) ∧ (𝑧 = 𝑡 ∧ 𝜑)))
22		anass 401	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 = 𝑟 ∧ 𝑦 = 𝑠) ∧ (𝑧 = 𝑡 ∧ 𝜑)) ↔ (𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))))
23	20, 21, 22	3bitri 206	. . . . . . . . . . . . . . . 16 ⊢ ((⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) ↔ (𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))))
24	23	3exbii 1631	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥∃𝑦∃𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))))
25		oprabidlem 5988	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥∃𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))) → ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))))
26	25	eximi 1624	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑦∃𝑥∃𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))) → ∃𝑦∃𝑥(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))))
27		excom 1688	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑥∃𝑦∃𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))) ↔ ∃𝑦∃𝑥∃𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))))
28		excom 1688	. . . . . . . . . . . . . . . . 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 201	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥∃𝑦∃𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))) → ∃𝑥∃𝑦(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))))
30		oprabidlem 5988	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥∃𝑦(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))) → ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))))
31		oprabidlem 5988	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑦∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑)) → ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑)))
32	31	anim2i 342	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑟 ∧ ∃𝑦∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡 ∧ 𝜑))) → (𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))))
33	32	eximi 1624	. . . . . . . . . . . . . . . 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 121	. . . . . . . . . . . . . 14 ⊢ (∃𝑥∃𝑦∃𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) → ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))))
36		euequ1 2150	. . . . . . . . . . . . . . . . . . 19 ⊢ ∃!𝑥 𝑥 = 𝑟
37		eupick 2134	. . . . . . . . . . . . . . . . . . 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 424	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))) → (𝑥 = 𝑟 → ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))))
39		euequ1 2150	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∃!𝑦 𝑦 = 𝑠
40		eupick 2134	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∃!𝑦 𝑦 = 𝑠 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))) → (𝑦 = 𝑠 → ∃𝑧(𝑧 = 𝑡 ∧ 𝜑)))
41	39, 40	mpan 424	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑)) → (𝑦 = 𝑠 → ∃𝑧(𝑧 = 𝑡 ∧ 𝜑)))
42		euequ1 2150	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∃!𝑧 𝑧 = 𝑡
43		eupick 2134	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∃!𝑧 𝑧 = 𝑡 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑)) → (𝑧 = 𝑡 → 𝜑))
44	42, 43	mpan 424	. . . . . . . . . . . . . . . . . . 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 1224	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡 ∧ 𝜑))) → ((𝑥 = 𝑟 ∧ 𝑦 = 𝑠 ∧ 𝑧 = 𝑡) → 𝜑))
48	17, 47	biimtrid 152	. . . . . . . . . . . . . . 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 2213	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
52		eqcom 2208	. . . . . . . . . . . . . . 15 ⊢ (⟨⟨𝑟, 𝑠⟩, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩)
53	51, 52	bitrdi 196	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩))
54	53	anbi1d 465	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑)))
55	54	3exbidv 1893	. . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 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 168	. . . . . . . . . . . 12 ⊢ (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑)))
59	16, 58	biimtrdi 163	. . . . . . . . . . 11 ⊢ (𝑎 = ⟨𝑟, 𝑠⟩ → (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))))
60	59	adantr 276	. . . . . . . . . 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 1921	. . . . . . . . 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 121	. . . . . . . 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 276	. . . . 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 1921	. . . 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 1614	. . . . 5 ⊢ ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
69		19.8a 1614	. . . . 5 ⊢ (∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
70		19.8a 1614	. . . . 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 115	. . 3 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑 → ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
73	67, 72	impbid 129	. 2 ⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ 𝜑))
74		df-oprab 5961	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
75	6, 73, 74	elab2 2925	1 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 981   = wceq 1373  ∃wex 1516  ∃!weu 2055   ∈ wcel 2177  Vcvv 2773  ⟨cop 3641  {coprab 5958

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-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-setind 4593

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-oprab 5961

This theorem is referenced by:  ssoprab2b  6015  ovid  6075  ovidig  6076  tposoprab  6379  xpcomco  6936