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Theorem oprabid 5537
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 constraint between 𝑥, 𝑦, and 𝑧, we use ax-bndl 1399 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.
StepHypRef Expression
1 vex 2560 . . . 4 𝑥 ∈ V
2 vex 2560 . . . 4 𝑦 ∈ V
31, 2opex 3966 . . 3 𝑥, 𝑦⟩ ∈ V
4 vex 2560 . . 3 𝑧 ∈ V
5 opexg 3964 . . 3 ((⟨𝑥, 𝑦⟩ ∈ V ∧ 𝑧 ∈ V) → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ V)
63, 4, 5mp2an 402 . 2 ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ V
73, 4eqvinop 3980 . . . . 5 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑎𝑡(𝑤 = ⟨𝑎, 𝑡⟩ ∧ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
87biimpi 113 . . . 4 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ∃𝑎𝑡(𝑤 = ⟨𝑎, 𝑡⟩ ∧ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
9 eqeq1 2046 . . . . . . . 8 (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
10 vex 2560 . . . . . . . . 9 𝑎 ∈ V
11 vex 2560 . . . . . . . . 9 𝑡 ∈ V
1210, 11opth1 3973 . . . . . . . 8 (⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑎 = ⟨𝑥, 𝑦⟩)
139, 12syl6bi 152 . . . . . . 7 (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑎 = ⟨𝑥, 𝑦⟩))
141, 2eqvinop 3980 . . . . . . . . 9 (𝑎 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑟𝑠(𝑎 = ⟨𝑟, 𝑠⟩ ∧ ⟨𝑟, 𝑠⟩ = ⟨𝑥, 𝑦⟩))
15 opeq1 3549 . . . . . . . . . . . . 13 (𝑎 = ⟨𝑟, 𝑠⟩ → ⟨𝑎, 𝑡⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩)
1615eqeq2d 2051 . . . . . . . . . . . 12 (𝑎 = ⟨𝑟, 𝑠⟩ → (𝑤 = ⟨𝑎, 𝑡⟩ ↔ 𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩))
171, 2, 4otth2 3978 . . . . . . . . . . . . . . . . . . 19 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ↔ (𝑥 = 𝑟𝑦 = 𝑠𝑧 = 𝑡))
18 df-3an 887 . . . . . . . . . . . . . . . . . . 19 ((𝑥 = 𝑟𝑦 = 𝑠𝑧 = 𝑡) ↔ ((𝑥 = 𝑟𝑦 = 𝑠) ∧ 𝑧 = 𝑡))
1917, 18bitri 173 . . . . . . . . . . . . . . . . . 18 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ↔ ((𝑥 = 𝑟𝑦 = 𝑠) ∧ 𝑧 = 𝑡))
2019anbi1i 431 . . . . . . . . . . . . . . . . 17 ((⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) ↔ (((𝑥 = 𝑟𝑦 = 𝑠) ∧ 𝑧 = 𝑡) ∧ 𝜑))
21 anass 381 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑟𝑦 = 𝑠) ∧ 𝑧 = 𝑡) ∧ 𝜑) ↔ ((𝑥 = 𝑟𝑦 = 𝑠) ∧ (𝑧 = 𝑡𝜑)))
22 anass 381 . . . . . . . . . . . . . . . . 17 (((𝑥 = 𝑟𝑦 = 𝑠) ∧ (𝑧 = 𝑡𝜑)) ↔ (𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
2320, 21, 223bitri 195 . . . . . . . . . . . . . . . 16 ((⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) ↔ (𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
24233exbii 1498 . . . . . . . . . . . . . . 15 (∃𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
25 oprabidlem 5536 . . . . . . . . . . . . . . . . . 18 (∃𝑥𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))) → ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
2625eximi 1491 . . . . . . . . . . . . . . . . 17 (∃𝑦𝑥𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))) → ∃𝑦𝑥(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
27 excom 1554 . . . . . . . . . . . . . . . . 17 (∃𝑥𝑦𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))) ↔ ∃𝑦𝑥𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
28 excom 1554 . . . . . . . . . . . . . . . . 17 (∃𝑥𝑦(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))) ↔ ∃𝑦𝑥(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
2926, 27, 283imtr4i 190 . . . . . . . . . . . . . . . 16 (∃𝑥𝑦𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))) → ∃𝑥𝑦(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
30 oprabidlem 5536 . . . . . . . . . . . . . . . 16 (∃𝑥𝑦(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))) → ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
31 oprabidlem 5536 . . . . . . . . . . . . . . . . . 18 (∃𝑦𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑)) → ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑)))
3231anim2i 324 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑟 ∧ ∃𝑦𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))) → (𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))))
3332eximi 1491 . . . . . . . . . . . . . . . 16 (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))) → ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))))
3429, 30, 333syl 17 . . . . . . . . . . . . . . 15 (∃𝑥𝑦𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))) → ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))))
3524, 34sylbi 114 . . . . . . . . . . . . . 14 (∃𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) → ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))))
36 euequ1 1995 . . . . . . . . . . . . . . . . . . 19 ∃!𝑥 𝑥 = 𝑟
37 eupick 1979 . . . . . . . . . . . . . . . . . . 19 ((∃!𝑥 𝑥 = 𝑟 ∧ ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑)))) → (𝑥 = 𝑟 → ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))))
3836, 37mpan 400 . . . . . . . . . . . . . . . . . 18 (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))) → (𝑥 = 𝑟 → ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))))
39 euequ1 1995 . . . . . . . . . . . . . . . . . . . 20 ∃!𝑦 𝑦 = 𝑠
40 eupick 1979 . . . . . . . . . . . . . . . . . . . 20 ((∃!𝑦 𝑦 = 𝑠 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))) → (𝑦 = 𝑠 → ∃𝑧(𝑧 = 𝑡𝜑)))
4139, 40mpan 400 . . . . . . . . . . . . . . . . . . 19 (∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑)) → (𝑦 = 𝑠 → ∃𝑧(𝑧 = 𝑡𝜑)))
42 euequ1 1995 . . . . . . . . . . . . . . . . . . . 20 ∃!𝑧 𝑧 = 𝑡
43 eupick 1979 . . . . . . . . . . . . . . . . . . . 20 ((∃!𝑧 𝑧 = 𝑡 ∧ ∃𝑧(𝑧 = 𝑡𝜑)) → (𝑧 = 𝑡𝜑))
4442, 43mpan 400 . . . . . . . . . . . . . . . . . . 19 (∃𝑧(𝑧 = 𝑡𝜑) → (𝑧 = 𝑡𝜑))
4541, 44syl6 29 . . . . . . . . . . . . . . . . . 18 (∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑)) → (𝑦 = 𝑠 → (𝑧 = 𝑡𝜑)))
4638, 45syl6 29 . . . . . . . . . . . . . . . . 17 (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))) → (𝑥 = 𝑟 → (𝑦 = 𝑠 → (𝑧 = 𝑡𝜑))))
47463impd 1118 . . . . . . . . . . . . . . . 16 (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))) → ((𝑥 = 𝑟𝑦 = 𝑠𝑧 = 𝑡) → 𝜑))
4817, 47syl5bi 141 . . . . . . . . . . . . . . 15 (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))) → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → 𝜑))
4948com12 27 . . . . . . . . . . . . . 14 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))) → 𝜑))
5035, 49syl5 28 . . . . . . . . . . . . 13 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (∃𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) → 𝜑))
51 eqeq1 2046 . . . . . . . . . . . . . . 15 (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
52 eqcom 2042 . . . . . . . . . . . . . . 15 (⟨⟨𝑟, 𝑠⟩, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩)
5351, 52syl6bb 185 . . . . . . . . . . . . . 14 (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩))
5453anbi1d 438 . . . . . . . . . . . . . . . 16 (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑)))
55543exbidv 1749 . . . . . . . . . . . . . . 15 (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑)))
5655imbi1d 220 . . . . . . . . . . . . . 14 (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → ((∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑) ↔ (∃𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) → 𝜑)))
5753, 56imbi12d 223 . . . . . . . . . . . . 13 (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑)) ↔ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (∃𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) → 𝜑))))
5850, 57mpbiri 157 . . . . . . . . . . . 12 (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑)))
5916, 58syl6bi 152 . . . . . . . . . . 11 (𝑎 = ⟨𝑟, 𝑠⟩ → (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))))
6059adantr 261 . . . . . . . . . 10 ((𝑎 = ⟨𝑟, 𝑠⟩ ∧ ⟨𝑟, 𝑠⟩ = ⟨𝑥, 𝑦⟩) → (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))))
6160exlimivv 1776 . . . . . . . . 9 (∃𝑟𝑠(𝑎 = ⟨𝑟, 𝑠⟩ ∧ ⟨𝑟, 𝑠⟩ = ⟨𝑥, 𝑦⟩) → (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))))
6214, 61sylbi 114 . . . . . . . 8 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))))
6362com3l 75 . . . . . . 7 (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑎 = ⟨𝑥, 𝑦⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))))
6413, 63mpdd 36 . . . . . 6 (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑)))
6564adantr 261 . . . . 5 ((𝑤 = ⟨𝑎, 𝑡⟩ ∧ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑)))
6665exlimivv 1776 . . . 4 (∃𝑎𝑡(𝑤 = ⟨𝑎, 𝑡⟩ ∧ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑)))
678, 66mpcom 32 . . 3 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))
68 19.8a 1482 . . . . 5 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
69 19.8a 1482 . . . . 5 (∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
70 19.8a 1482 . . . . 5 (∃𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
7168, 69, 703syl 17 . . . 4 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
7271ex 108 . . 3 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑 → ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
7367, 72impbid 120 . 2 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ 𝜑))
74 df-oprab 5516 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
756, 73, 74elab2 2690 1 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  w3a 885   = wceq 1243  wex 1381  wcel 1393  ∃!weu 1900  Vcvv 2557  cop 3378  {coprab 5513
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-oprab 5516
This theorem is referenced by:  ssoprab2b  5562  ovid  5617  ovidig  5618  tposoprab  5895  xpcomco  6300
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