Proof of Theorem reu8
Step | Hyp | Ref
| Expression |
1 | | rmo4.1 |
. . 3
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
2 | 1 | cbvreuvw 3351 |
. 2
⊢
(∃!𝑥 ∈
𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) |
3 | | reu6 3628 |
. 2
⊢
(∃!𝑦 ∈
𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥)) |
4 | | dfbi2 478 |
. . . . 5
⊢ ((𝜓 ↔ 𝑦 = 𝑥) ↔ ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓))) |
5 | 4 | ralbii 3078 |
. . . 4
⊢
(∀𝑦 ∈
𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ ∀𝑦 ∈ 𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓))) |
6 | | r19.26 3082 |
. . . . 5
⊢
(∀𝑦 ∈
𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓))) |
7 | | ancom 464 |
. . . . . 6
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ∧ 𝜑)) |
8 | | equcom 2028 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) |
9 | 8 | imbi2i 339 |
. . . . . . . . 9
⊢ ((𝜓 → 𝑥 = 𝑦) ↔ (𝜓 → 𝑦 = 𝑥)) |
10 | 9 | ralbii 3078 |
. . . . . . . 8
⊢
(∀𝑦 ∈
𝐴 (𝜓 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥)) |
11 | 10 | a1i 11 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥))) |
12 | | biimt 364 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 → 𝜑))) |
13 | | df-ral 3056 |
. . . . . . . . 9
⊢
(∀𝑦 ∈
𝐴 (𝑦 = 𝑥 → 𝜓) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓))) |
14 | | bi2.04 392 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓)) ↔ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓))) |
15 | 14 | albii 1827 |
. . . . . . . . 9
⊢
(∀𝑦(𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓))) |
16 | | eleq1w 2813 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) |
17 | 16, 1 | imbi12d 348 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜓))) |
18 | 17 | bicomd 226 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → ((𝑦 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜑))) |
19 | 18 | equcoms 2030 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜑))) |
20 | 19 | equsalvw 2013 |
. . . . . . . . 9
⊢
(∀𝑦(𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓)) ↔ (𝑥 ∈ 𝐴 → 𝜑)) |
21 | 13, 15, 20 | 3bitrri 301 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)) |
22 | 12, 21 | bitrdi 290 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓))) |
23 | 11, 22 | anbi12d 634 |
. . . . . 6
⊢ (𝑥 ∈ 𝐴 → ((∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ∧ 𝜑) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)))) |
24 | 7, 23 | syl5bb 286 |
. . . . 5
⊢ (𝑥 ∈ 𝐴 → ((𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)))) |
25 | 6, 24 | bitr4id 293 |
. . . 4
⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))) |
26 | 5, 25 | syl5bb 286 |
. . 3
⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))) |
27 | 26 | rexbiia 3159 |
. 2
⊢
(∃𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
28 | 2, 3, 27 | 3bitri 300 |
1
⊢
(∃!𝑥 ∈
𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |