Proof of Theorem imaeqalov
Step | Hyp | Ref
| Expression |
1 | | df-ral 3061 |
. . 3
⊢
(∀𝑥 ∈
(𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) → 𝜑)) |
2 | | ovelimab 7567 |
. . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧))) |
3 | 2 | imbi1d 341 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → ((𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) → 𝜑) ↔ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))) |
4 | 3 | albidv 1923 |
. . 3
⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∀𝑥(𝑥 ∈ (𝐹 “ (𝐵 × 𝐶)) → 𝜑) ↔ ∀𝑥(∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))) |
5 | 1, 4 | bitrid 282 |
. 2
⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑥(∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑))) |
6 | | ralcom4 3282 |
. . . 4
⊢
(∀𝑦 ∈
𝐵 ∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑)) |
7 | | r19.23v 3181 |
. . . . . . 7
⊢
(∀𝑧 ∈
𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ (∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑)) |
8 | 7 | ralbii 3092 |
. . . . . 6
⊢
(∀𝑦 ∈
𝐵 ∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑦 ∈ 𝐵 (∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑)) |
9 | | r19.23v 3181 |
. . . . . 6
⊢
(∀𝑦 ∈
𝐵 (∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑)) |
10 | 8, 9 | bitri 274 |
. . . . 5
⊢
(∀𝑦 ∈
𝐵 ∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑)) |
11 | 10 | albii 1821 |
. . . 4
⊢
(∀𝑥∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥(∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑)) |
12 | 6, 11 | bitri 274 |
. . 3
⊢
(∀𝑦 ∈
𝐵 ∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥(∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑)) |
13 | | ralcom4 3282 |
. . . . 5
⊢
(∀𝑧 ∈
𝐶 ∀𝑥(𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑)) |
14 | | ovex 7425 |
. . . . . . 7
⊢ (𝑦𝐹𝑧) ∈ V |
15 | | imaeqexov.1 |
. . . . . . 7
⊢ (𝑥 = (𝑦𝐹𝑧) → (𝜑 ↔ 𝜓)) |
16 | 14, 15 | ceqsalv 3508 |
. . . . . 6
⊢
(∀𝑥(𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ 𝜓) |
17 | 16 | ralbii 3092 |
. . . . 5
⊢
(∀𝑧 ∈
𝐶 ∀𝑥(𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑧 ∈ 𝐶 𝜓) |
18 | 13, 17 | bitr3i 276 |
. . . 4
⊢
(∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑧 ∈ 𝐶 𝜓) |
19 | 18 | ralbii 3092 |
. . 3
⊢
(∀𝑦 ∈
𝐵 ∀𝑥∀𝑧 ∈ 𝐶 (𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓) |
20 | 12, 19 | bitr3i 276 |
. 2
⊢
(∀𝑥(∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦𝐹𝑧) → 𝜑) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓) |
21 | 5, 20 | bitrdi 286 |
1
⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)) |