Step | Hyp | Ref
| Expression |
1 | | nfv 1922 |
. . . 4
⊢
Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝜑) |
2 | | nfcsb1v 3836 |
. . . . . 6
⊢
Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴 |
3 | 2 | nfcri 2891 |
. . . . 5
⊢
Ⅎ𝑥 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 |
4 | | nfs1v 2157 |
. . . . 5
⊢
Ⅎ𝑥[𝑧 / 𝑥]𝜑 |
5 | 3, 4 | nfan 1907 |
. . . 4
⊢
Ⅎ𝑥(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑) |
6 | | id 22 |
. . . . . 6
⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧) |
7 | | csbeq1a 3825 |
. . . . . 6
⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴) |
8 | 6, 7 | eleq12d 2832 |
. . . . 5
⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴)) |
9 | | sbequ12 2249 |
. . . . 5
⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) |
10 | 8, 9 | anbi12d 634 |
. . . 4
⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑))) |
11 | 1, 5, 10 | cbveu 2608 |
. . 3
⊢
(∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑)) |
12 | | nfcv 2904 |
. . . . . . 7
⊢
Ⅎ𝑦𝑧 |
13 | | cbvralcsf.1 |
. . . . . . 7
⊢
Ⅎ𝑦𝐴 |
14 | 12, 13 | nfcsb 3839 |
. . . . . 6
⊢
Ⅎ𝑦⦋𝑧 / 𝑥⦌𝐴 |
15 | 14 | nfcri 2891 |
. . . . 5
⊢
Ⅎ𝑦 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 |
16 | | cbvralcsf.3 |
. . . . . 6
⊢
Ⅎ𝑦𝜑 |
17 | 16 | nfsb 2526 |
. . . . 5
⊢
Ⅎ𝑦[𝑧 / 𝑥]𝜑 |
18 | 15, 17 | nfan 1907 |
. . . 4
⊢
Ⅎ𝑦(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑) |
19 | | nfv 1922 |
. . . 4
⊢
Ⅎ𝑧(𝑦 ∈ 𝐵 ∧ 𝜓) |
20 | | id 22 |
. . . . . 6
⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦) |
21 | | csbeq1 3814 |
. . . . . . 7
⊢ (𝑧 = 𝑦 → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐴) |
22 | | sbsbc 3698 |
. . . . . . . . 9
⊢ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑣 ∈ 𝐴) |
23 | 22 | abbii 2808 |
. . . . . . . 8
⊢ {𝑣 ∣ [𝑦 / 𝑥]𝑣 ∈ 𝐴} = {𝑣 ∣ [𝑦 / 𝑥]𝑣 ∈ 𝐴} |
24 | | cbvralcsf.2 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥𝐵 |
25 | 24 | nfcri 2891 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥 𝑣 ∈ 𝐵 |
26 | | cbvralcsf.5 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
27 | 26 | eleq2d 2823 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑣 ∈ 𝐴 ↔ 𝑣 ∈ 𝐵)) |
28 | 25, 27 | sbie 2505 |
. . . . . . . . . 10
⊢ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ↔ 𝑣 ∈ 𝐵) |
29 | 28 | bicomi 227 |
. . . . . . . . 9
⊢ (𝑣 ∈ 𝐵 ↔ [𝑦 / 𝑥]𝑣 ∈ 𝐴) |
30 | 29 | abbi2i 2876 |
. . . . . . . 8
⊢ 𝐵 = {𝑣 ∣ [𝑦 / 𝑥]𝑣 ∈ 𝐴} |
31 | | df-csb 3812 |
. . . . . . . 8
⊢
⦋𝑦 /
𝑥⦌𝐴 = {𝑣 ∣ [𝑦 / 𝑥]𝑣 ∈ 𝐴} |
32 | 23, 30, 31 | 3eqtr4ri 2776 |
. . . . . . 7
⊢
⦋𝑦 /
𝑥⦌𝐴 = 𝐵 |
33 | 21, 32 | eqtrdi 2794 |
. . . . . 6
⊢ (𝑧 = 𝑦 → ⦋𝑧 / 𝑥⦌𝐴 = 𝐵) |
34 | 20, 33 | eleq12d 2832 |
. . . . 5
⊢ (𝑧 = 𝑦 → (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ↔ 𝑦 ∈ 𝐵)) |
35 | | sbequ 2089 |
. . . . . 6
⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
36 | | cbvralcsf.4 |
. . . . . . 7
⊢
Ⅎ𝑥𝜓 |
37 | | cbvralcsf.6 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
38 | 36, 37 | sbie 2505 |
. . . . . 6
⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
39 | 35, 38 | bitrdi 290 |
. . . . 5
⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓)) |
40 | 34, 39 | anbi12d 634 |
. . . 4
⊢ (𝑧 = 𝑦 → ((𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐵 ∧ 𝜓))) |
41 | 18, 19, 40 | cbveu 2608 |
. . 3
⊢
(∃!𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝜓)) |
42 | 11, 41 | bitri 278 |
. 2
⊢
(∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝜓)) |
43 | | df-reu 3068 |
. 2
⊢
(∃!𝑥 ∈
𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
44 | | df-reu 3068 |
. 2
⊢
(∃!𝑦 ∈
𝐵 𝜓 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝜓)) |
45 | 42, 43, 44 | 3bitr4i 306 |
1
⊢
(∃!𝑥 ∈
𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐵 𝜓) |