Proof of Theorem raaan2
| Step | Hyp | Ref
| Expression |
| 1 | | dfbi3 1050 |
. 2
⊢ ((𝐴 = ∅ ↔ 𝐵 = ∅) ↔ ((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))) |
| 2 | | rzal 4509 |
. . . . 5
⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓)) |
| 3 | 2 | adantr 480 |
. . . 4
⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓)) |
| 4 | | rzal 4509 |
. . . . 5
⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
| 5 | 4 | adantr 480 |
. . . 4
⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑥 ∈ 𝐴 𝜑) |
| 6 | | rzal 4509 |
. . . . 5
⊢ (𝐵 = ∅ → ∀𝑦 ∈ 𝐵 𝜓) |
| 7 | 6 | adantl 481 |
. . . 4
⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑦 ∈ 𝐵 𝜓) |
| 8 | | pm5.1 824 |
. . . 4
⊢
((∀𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))) |
| 9 | 3, 5, 7, 8 | syl12anc 837 |
. . 3
⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) →
(∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))) |
| 10 | | df-ne 2941 |
. . . . 5
⊢ (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅) |
| 11 | | raaan2.1 |
. . . . . . 7
⊢
Ⅎ𝑦𝜑 |
| 12 | 11 | r19.28z 4498 |
. . . . . 6
⊢ (𝐵 ≠ ∅ →
(∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))) |
| 13 | 12 | ralbidv 3178 |
. . . . 5
⊢ (𝐵 ≠ ∅ →
(∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))) |
| 14 | 10, 13 | sylbir 235 |
. . . 4
⊢ (¬
𝐵 = ∅ →
(∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))) |
| 15 | | df-ne 2941 |
. . . . 5
⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) |
| 16 | | nfcv 2905 |
. . . . . . 7
⊢
Ⅎ𝑥𝐵 |
| 17 | | raaan2.2 |
. . . . . . 7
⊢
Ⅎ𝑥𝜓 |
| 18 | 16, 17 | nfralw 3311 |
. . . . . 6
⊢
Ⅎ𝑥∀𝑦 ∈ 𝐵 𝜓 |
| 19 | 18 | r19.27z 4505 |
. . . . 5
⊢ (𝐴 ≠ ∅ →
(∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))) |
| 20 | 15, 19 | sylbir 235 |
. . . 4
⊢ (¬
𝐴 = ∅ →
(∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))) |
| 21 | 14, 20 | sylan9bbr 510 |
. . 3
⊢ ((¬
𝐴 = ∅ ∧ ¬
𝐵 = ∅) →
(∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))) |
| 22 | 9, 21 | jaoi 858 |
. 2
⊢ (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)) →
(∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))) |
| 23 | 1, 22 | sylbi 217 |
1
⊢ ((𝐴 = ∅ ↔ 𝐵 = ∅) →
(∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))) |