Step | Hyp | Ref
| Expression |
1 | | rabn0 4319 |
. 2
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝜑) |
2 | | frminex.1 |
. . . . 5
⊢ 𝐴 ∈ V |
3 | 2 | rabex 5256 |
. . . 4
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V |
4 | | ssrab2 4013 |
. . . 4
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 |
5 | | fri 5549 |
. . . . . 6
⊢ ((({𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅)) → ∃𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ¬ 𝑦𝑅𝑧) |
6 | | frminex.2 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
7 | 6 | ralrab 3630 |
. . . . . . . 8
⊢
(∀𝑦 ∈
{𝑥 ∈ 𝐴 ∣ 𝜑} ¬ 𝑦𝑅𝑧 ↔ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑧)) |
8 | 7 | rexbii 3181 |
. . . . . . 7
⊢
(∃𝑧 ∈
{𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ¬ 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑧)) |
9 | | breq2 5078 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑥 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝑥)) |
10 | 9 | notbid 318 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑦𝑅𝑥)) |
11 | 10 | imbi2d 341 |
. . . . . . . . 9
⊢ (𝑧 = 𝑥 → ((𝜓 → ¬ 𝑦𝑅𝑧) ↔ (𝜓 → ¬ 𝑦𝑅𝑥))) |
12 | 11 | ralbidv 3112 |
. . . . . . . 8
⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑧) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))) |
13 | 12 | rexrab2 3637 |
. . . . . . 7
⊢
(∃𝑧 ∈
{𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑧) ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))) |
14 | 8, 13 | bitri 274 |
. . . . . 6
⊢
(∃𝑧 ∈
{𝑥 ∈ 𝐴 ∣ 𝜑}∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ¬ 𝑦𝑅𝑧 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))) |
15 | 5, 14 | sylib 217 |
. . . . 5
⊢ ((({𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅)) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))) |
16 | 15 | an4s 657 |
. . . 4
⊢ ((({𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V ∧ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴) ∧ (𝑅 Fr 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅)) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))) |
17 | 3, 4, 16 | mpanl12 699 |
. . 3
⊢ ((𝑅 Fr 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))) |
18 | 17 | ex 413 |
. 2
⊢ (𝑅 Fr 𝐴 → ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))) |
19 | 1, 18 | syl5bir 242 |
1
⊢ (𝑅 Fr 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))) |