Proof of Theorem reu8
Step | Hyp | Ref
| Expression |
1 | | rmo4.1 |
. . 3
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
2 | 1 | cbvreuv 3405 |
. 2
⊢
(∃!𝑥 ∈
𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) |
3 | | reu6 3651 |
. 2
⊢
(∃!𝑦 ∈
𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥)) |
4 | | dfbi2 475 |
. . . . 5
⊢ ((𝜓 ↔ 𝑦 = 𝑥) ↔ ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓))) |
5 | 4 | ralbii 3132 |
. . . 4
⊢
(∀𝑦 ∈
𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ ∀𝑦 ∈ 𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓))) |
6 | | ancom 461 |
. . . . . 6
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ∧ 𝜑)) |
7 | | equcom 2002 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) |
8 | 7 | imbi2i 337 |
. . . . . . . . 9
⊢ ((𝜓 → 𝑥 = 𝑦) ↔ (𝜓 → 𝑦 = 𝑥)) |
9 | 8 | ralbii 3132 |
. . . . . . . 8
⊢
(∀𝑦 ∈
𝐴 (𝜓 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥)) |
10 | 9 | a1i 11 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥))) |
11 | | biimt 362 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 → 𝜑))) |
12 | | df-ral 3110 |
. . . . . . . . 9
⊢
(∀𝑦 ∈
𝐴 (𝑦 = 𝑥 → 𝜓) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓))) |
13 | | bi2.04 389 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓)) ↔ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓))) |
14 | 13 | albii 1801 |
. . . . . . . . 9
⊢
(∀𝑦(𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓))) |
15 | | eleq1w 2865 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) |
16 | 15, 1 | imbi12d 346 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜓))) |
17 | 16 | bicomd 224 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → ((𝑦 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜑))) |
18 | 17 | equcoms 2004 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜑))) |
19 | 18 | equsalvw 1987 |
. . . . . . . . 9
⊢
(∀𝑦(𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓)) ↔ (𝑥 ∈ 𝐴 → 𝜑)) |
20 | 12, 14, 19 | 3bitrri 299 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)) |
21 | 11, 20 | syl6bb 288 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓))) |
22 | 10, 21 | anbi12d 630 |
. . . . . 6
⊢ (𝑥 ∈ 𝐴 → ((∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ∧ 𝜑) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)))) |
23 | 6, 22 | syl5bb 284 |
. . . . 5
⊢ (𝑥 ∈ 𝐴 → ((𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)))) |
24 | | r19.26 3137 |
. . . . 5
⊢
(∀𝑦 ∈
𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓))) |
25 | 23, 24 | syl6rbbr 291 |
. . . 4
⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))) |
26 | 5, 25 | syl5bb 284 |
. . 3
⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))) |
27 | 26 | rexbiia 3210 |
. 2
⊢
(∃𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
28 | 2, 3, 27 | 3bitri 298 |
1
⊢
(∃!𝑥 ∈
𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |