Proof of Theorem reu8
Step | Hyp | Ref
| Expression |
1 | | rmo4.1 |
. . 3
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
2 | 1 | cbvreuv 2698 |
. 2
⊢
(∃!𝑥 ∈
𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) |
3 | | reu6 2919 |
. 2
⊢
(∃!𝑦 ∈
𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥)) |
4 | | dfbi2 386 |
. . . . 5
⊢ ((𝜓 ↔ 𝑦 = 𝑥) ↔ ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓))) |
5 | 4 | ralbii 2476 |
. . . 4
⊢
(∀𝑦 ∈
𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ ∀𝑦 ∈ 𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓))) |
6 | | r19.26 2596 |
. . . . 5
⊢
(∀𝑦 ∈
𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓))) |
7 | | ancom 264 |
. . . . . 6
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ∧ 𝜑)) |
8 | | equcom 1699 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) |
9 | 8 | imbi2i 225 |
. . . . . . . . 9
⊢ ((𝜓 → 𝑥 = 𝑦) ↔ (𝜓 → 𝑦 = 𝑥)) |
10 | 9 | ralbii 2476 |
. . . . . . . 8
⊢
(∀𝑦 ∈
𝐴 (𝜓 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥)) |
11 | 10 | a1i 9 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥))) |
12 | | biimt 240 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 → 𝜑))) |
13 | | df-ral 2453 |
. . . . . . . . 9
⊢
(∀𝑦 ∈
𝐴 (𝑦 = 𝑥 → 𝜓) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓))) |
14 | | bi2.04 247 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓)) ↔ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓))) |
15 | 14 | albii 1463 |
. . . . . . . . 9
⊢
(∀𝑦(𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓))) |
16 | | vex 2733 |
. . . . . . . . . 10
⊢ 𝑥 ∈ V |
17 | | eleq1 2233 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) |
18 | 17, 1 | imbi12d 233 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜓))) |
19 | 18 | bicomd 140 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → ((𝑦 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜑))) |
20 | 19 | equcoms 1701 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜑))) |
21 | 16, 20 | ceqsalv 2760 |
. . . . . . . . 9
⊢
(∀𝑦(𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓)) ↔ (𝑥 ∈ 𝐴 → 𝜑)) |
22 | 13, 15, 21 | 3bitrri 206 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)) |
23 | 12, 22 | bitrdi 195 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓))) |
24 | 11, 23 | anbi12d 470 |
. . . . . 6
⊢ (𝑥 ∈ 𝐴 → ((∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ∧ 𝜑) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)))) |
25 | 7, 24 | syl5bb 191 |
. . . . 5
⊢ (𝑥 ∈ 𝐴 → ((𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)))) |
26 | 6, 25 | bitr4id 198 |
. . . 4
⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))) |
27 | 5, 26 | syl5bb 191 |
. . 3
⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))) |
28 | 27 | rexbiia 2485 |
. 2
⊢
(∃𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
29 | 2, 3, 28 | 3bitri 205 |
1
⊢
(∃!𝑥 ∈
𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |