ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  oprabid GIF version

Theorem oprabid 5638
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 1442 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 2618 . . . 4 𝑥 ∈ V
2 vex 2618 . . . 4 𝑦 ∈ V
31, 2opex 4030 . . 3 𝑥, 𝑦⟩ ∈ V
4 vex 2618 . . 3 𝑧 ∈ V
5 opexg 4029 . . 3 ((⟨𝑥, 𝑦⟩ ∈ V ∧ 𝑧 ∈ V) → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ V)
63, 4, 5mp2an 417 . 2 ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ V
73, 4eqvinop 4044 . . . . 5 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑎𝑡(𝑤 = ⟨𝑎, 𝑡⟩ ∧ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
87biimpi 118 . . . 4 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ∃𝑎𝑡(𝑤 = ⟨𝑎, 𝑡⟩ ∧ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
9 eqeq1 2091 . . . . . . . 8 (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
10 vex 2618 . . . . . . . . 9 𝑎 ∈ V
11 vex 2618 . . . . . . . . 9 𝑡 ∈ V
1210, 11opth1 4037 . . . . . . . 8 (⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑎 = ⟨𝑥, 𝑦⟩)
139, 12syl6bi 161 . . . . . . 7 (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑎 = ⟨𝑥, 𝑦⟩))
141, 2eqvinop 4044 . . . . . . . . 9 (𝑎 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑟𝑠(𝑎 = ⟨𝑟, 𝑠⟩ ∧ ⟨𝑟, 𝑠⟩ = ⟨𝑥, 𝑦⟩))
15 opeq1 3605 . . . . . . . . . . . . 13 (𝑎 = ⟨𝑟, 𝑠⟩ → ⟨𝑎, 𝑡⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩)
1615eqeq2d 2096 . . . . . . . . . . . 12 (𝑎 = ⟨𝑟, 𝑠⟩ → (𝑤 = ⟨𝑎, 𝑡⟩ ↔ 𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩))
171, 2, 4otth2 4042 . . . . . . . . . . . . . . . . . . 19 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ↔ (𝑥 = 𝑟𝑦 = 𝑠𝑧 = 𝑡))
18 df-3an 924 . . . . . . . . . . . . . . . . . . 19 ((𝑥 = 𝑟𝑦 = 𝑠𝑧 = 𝑡) ↔ ((𝑥 = 𝑟𝑦 = 𝑠) ∧ 𝑧 = 𝑡))
1917, 18bitri 182 . . . . . . . . . . . . . . . . . 18 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ↔ ((𝑥 = 𝑟𝑦 = 𝑠) ∧ 𝑧 = 𝑡))
2019anbi1i 446 . . . . . . . . . . . . . . . . 17 ((⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) ↔ (((𝑥 = 𝑟𝑦 = 𝑠) ∧ 𝑧 = 𝑡) ∧ 𝜑))
21 anass 393 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑟𝑦 = 𝑠) ∧ 𝑧 = 𝑡) ∧ 𝜑) ↔ ((𝑥 = 𝑟𝑦 = 𝑠) ∧ (𝑧 = 𝑡𝜑)))
22 anass 393 . . . . . . . . . . . . . . . . 17 (((𝑥 = 𝑟𝑦 = 𝑠) ∧ (𝑧 = 𝑡𝜑)) ↔ (𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
2320, 21, 223bitri 204 . . . . . . . . . . . . . . . 16 ((⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) ↔ (𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
24233exbii 1541 . . . . . . . . . . . . . . 15 (∃𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
25 oprabidlem 5637 . . . . . . . . . . . . . . . . . 18 (∃𝑥𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))) → ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
2625eximi 1534 . . . . . . . . . . . . . . . . 17 (∃𝑦𝑥𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))) → ∃𝑦𝑥(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
27 excom 1597 . . . . . . . . . . . . . . . . 17 (∃𝑥𝑦𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))) ↔ ∃𝑦𝑥𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
28 excom 1597 . . . . . . . . . . . . . . . . 17 (∃𝑥𝑦(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))) ↔ ∃𝑦𝑥(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
2926, 27, 283imtr4i 199 . . . . . . . . . . . . . . . 16 (∃𝑥𝑦𝑧(𝑥 = 𝑟 ∧ (𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))) → ∃𝑥𝑦(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
30 oprabidlem 5637 . . . . . . . . . . . . . . . 16 (∃𝑥𝑦(𝑥 = 𝑟 ∧ ∃𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))) → ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))))
31 oprabidlem 5637 . . . . . . . . . . . . . . . . . 18 (∃𝑦𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑)) → ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑)))
3231anim2i 334 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑟 ∧ ∃𝑦𝑧(𝑦 = 𝑠 ∧ (𝑧 = 𝑡𝜑))) → (𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))))
3332eximi 1534 . . . . . . . . . . . . . . . 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 119 . . . . . . . . . . . . . 14 (∃𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) → ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))))
36 euequ1 2040 . . . . . . . . . . . . . . . . . . 19 ∃!𝑥 𝑥 = 𝑟
37 eupick 2024 . . . . . . . . . . . . . . . . . . 19 ((∃!𝑥 𝑥 = 𝑟 ∧ ∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑)))) → (𝑥 = 𝑟 → ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))))
3836, 37mpan 415 . . . . . . . . . . . . . . . . . 18 (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))) → (𝑥 = 𝑟 → ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))))
39 euequ1 2040 . . . . . . . . . . . . . . . . . . . 20 ∃!𝑦 𝑦 = 𝑠
40 eupick 2024 . . . . . . . . . . . . . . . . . . . 20 ((∃!𝑦 𝑦 = 𝑠 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))) → (𝑦 = 𝑠 → ∃𝑧(𝑧 = 𝑡𝜑)))
4139, 40mpan 415 . . . . . . . . . . . . . . . . . . 19 (∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑)) → (𝑦 = 𝑠 → ∃𝑧(𝑧 = 𝑡𝜑)))
42 euequ1 2040 . . . . . . . . . . . . . . . . . . . 20 ∃!𝑧 𝑧 = 𝑡
43 eupick 2024 . . . . . . . . . . . . . . . . . . . 20 ((∃!𝑧 𝑧 = 𝑡 ∧ ∃𝑧(𝑧 = 𝑡𝜑)) → (𝑧 = 𝑡𝜑))
4442, 43mpan 415 . . . . . . . . . . . . . . . . . . 19 (∃𝑧(𝑧 = 𝑡𝜑) → (𝑧 = 𝑡𝜑))
4541, 44syl6 33 . . . . . . . . . . . . . . . . . 18 (∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑)) → (𝑦 = 𝑠 → (𝑧 = 𝑡𝜑)))
4638, 45syl6 33 . . . . . . . . . . . . . . . . 17 (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))) → (𝑥 = 𝑟 → (𝑦 = 𝑠 → (𝑧 = 𝑡𝜑))))
47463impd 1155 . . . . . . . . . . . . . . . 16 (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))) → ((𝑥 = 𝑟𝑦 = 𝑠𝑧 = 𝑡) → 𝜑))
4817, 47syl5bi 150 . . . . . . . . . . . . . . 15 (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))) → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → 𝜑))
4948com12 30 . . . . . . . . . . . . . 14 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (∃𝑥(𝑥 = 𝑟 ∧ ∃𝑦(𝑦 = 𝑠 ∧ ∃𝑧(𝑧 = 𝑡𝜑))) → 𝜑))
5035, 49syl5 32 . . . . . . . . . . . . 13 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (∃𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) → 𝜑))
51 eqeq1 2091 . . . . . . . . . . . . . . 15 (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
52 eqcom 2087 . . . . . . . . . . . . . . 15 (⟨⟨𝑟, 𝑠⟩, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩)
5351, 52syl6bb 194 . . . . . . . . . . . . . 14 (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩))
5453anbi1d 453 . . . . . . . . . . . . . . . 16 (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑)))
55543exbidv 1794 . . . . . . . . . . . . . . 15 (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑)))
5655imbi1d 229 . . . . . . . . . . . . . 14 (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → ((∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑) ↔ (∃𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ ∧ 𝜑) → 𝜑)))
5753, 56imbi12d 232 . . . . . . . . . . . . 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 166 . . . . . . . . . . . 12 (𝑤 = ⟨⟨𝑟, 𝑠⟩, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑)))
5916, 58syl6bi 161 . . . . . . . . . . 11 (𝑎 = ⟨𝑟, 𝑠⟩ → (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))))
6059adantr 270 . . . . . . . . . 10 ((𝑎 = ⟨𝑟, 𝑠⟩ ∧ ⟨𝑟, 𝑠⟩ = ⟨𝑥, 𝑦⟩) → (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))))
6160exlimivv 1821 . . . . . . . . 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 119 . . . . . . . 8 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))))
6362com3l 80 . . . . . . 7 (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑎 = ⟨𝑥, 𝑦⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))))
6413, 63mpdd 40 . . . . . 6 (𝑤 = ⟨𝑎, 𝑡⟩ → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑)))
6564adantr 270 . . . . 5 ((𝑤 = ⟨𝑎, 𝑡⟩ ∧ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑)))
6665exlimivv 1821 . . . 4 (∃𝑎𝑡(𝑤 = ⟨𝑎, 𝑡⟩ ∧ ⟨𝑎, 𝑡⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑)))
678, 66mpcom 36 . . 3 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜑))
68 19.8a 1525 . . . . 5 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
69 19.8a 1525 . . . . 5 (∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
70 19.8a 1525 . . . . 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 113 . . 3 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑 → ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
7367, 72impbid 127 . 2 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ 𝜑))
74 df-oprab 5617 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
756, 73, 74elab2 2754 1 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 922   = wceq 1287  wex 1424  wcel 1436  ∃!weu 1945  Vcvv 2615  cop 3434  {coprab 5614
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-setind 4326
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-oprab 5617
This theorem is referenced by:  ssoprab2b  5663  ovid  5718  ovidig  5719  tposoprab  5999  xpcomco  6494
  Copyright terms: Public domain W3C validator