Step | Hyp | Ref
| Expression |
1 | | nfv 1508 |
. . . 4
⊢
Ⅎ𝑧(𝑥 ∈ 𝐴 → 𝜑) |
2 | | nfcsb1v 3064 |
. . . . . 6
⊢
Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴 |
3 | 2 | nfcri 2293 |
. . . . 5
⊢
Ⅎ𝑥 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 |
4 | | nfsbc1v 2955 |
. . . . 5
⊢
Ⅎ𝑥[𝑧 / 𝑥]𝜑 |
5 | 3, 4 | nfim 1552 |
. . . 4
⊢
Ⅎ𝑥(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 → [𝑧 / 𝑥]𝜑) |
6 | | id 19 |
. . . . . 6
⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧) |
7 | | csbeq1a 3040 |
. . . . . 6
⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴) |
8 | 6, 7 | eleq12d 2228 |
. . . . 5
⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴)) |
9 | | sbceq1a 2946 |
. . . . 5
⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) |
10 | 8, 9 | imbi12d 233 |
. . . 4
⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 → [𝑧 / 𝑥]𝜑))) |
11 | 1, 5, 10 | cbval 1734 |
. . 3
⊢
(∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 → [𝑧 / 𝑥]𝜑)) |
12 | | nfcv 2299 |
. . . . . . 7
⊢
Ⅎ𝑦𝑧 |
13 | | cbvralcsf.1 |
. . . . . . 7
⊢
Ⅎ𝑦𝐴 |
14 | 12, 13 | nfcsb 3068 |
. . . . . 6
⊢
Ⅎ𝑦⦋𝑧 / 𝑥⦌𝐴 |
15 | 14 | nfcri 2293 |
. . . . 5
⊢
Ⅎ𝑦 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 |
16 | | cbvralcsf.3 |
. . . . . 6
⊢
Ⅎ𝑦𝜑 |
17 | 12, 16 | nfsbc 2957 |
. . . . 5
⊢
Ⅎ𝑦[𝑧 / 𝑥]𝜑 |
18 | 15, 17 | nfim 1552 |
. . . 4
⊢
Ⅎ𝑦(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 → [𝑧 / 𝑥]𝜑) |
19 | | nfv 1508 |
. . . 4
⊢
Ⅎ𝑧(𝑦 ∈ 𝐵 → 𝜓) |
20 | | id 19 |
. . . . . 6
⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦) |
21 | | csbeq1 3034 |
. . . . . . 7
⊢ (𝑧 = 𝑦 → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐴) |
22 | | df-csb 3032 |
. . . . . . . 8
⊢
⦋𝑦 /
𝑥⦌𝐴 = {𝑣 ∣ [𝑦 / 𝑥]𝑣 ∈ 𝐴} |
23 | | cbvralcsf.2 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥𝐵 |
24 | 23 | nfcri 2293 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥 𝑣 ∈ 𝐵 |
25 | | cbvralcsf.5 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
26 | 25 | eleq2d 2227 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑣 ∈ 𝐴 ↔ 𝑣 ∈ 𝐵)) |
27 | 24, 26 | sbie 1771 |
. . . . . . . . . 10
⊢ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ↔ 𝑣 ∈ 𝐵) |
28 | | sbsbc 2941 |
. . . . . . . . . 10
⊢ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑣 ∈ 𝐴) |
29 | 27, 28 | bitr3i 185 |
. . . . . . . . 9
⊢ (𝑣 ∈ 𝐵 ↔ [𝑦 / 𝑥]𝑣 ∈ 𝐴) |
30 | 29 | abbi2i 2272 |
. . . . . . . 8
⊢ 𝐵 = {𝑣 ∣ [𝑦 / 𝑥]𝑣 ∈ 𝐴} |
31 | 22, 30 | eqtr4i 2181 |
. . . . . . 7
⊢
⦋𝑦 /
𝑥⦌𝐴 = 𝐵 |
32 | 21, 31 | eqtrdi 2206 |
. . . . . 6
⊢ (𝑧 = 𝑦 → ⦋𝑧 / 𝑥⦌𝐴 = 𝐵) |
33 | 20, 32 | eleq12d 2228 |
. . . . 5
⊢ (𝑧 = 𝑦 → (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ↔ 𝑦 ∈ 𝐵)) |
34 | | dfsbcq 2939 |
. . . . . 6
⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
35 | | sbsbc 2941 |
. . . . . . 7
⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
36 | | cbvralcsf.4 |
. . . . . . . 8
⊢
Ⅎ𝑥𝜓 |
37 | | cbvralcsf.6 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
38 | 36, 37 | sbie 1771 |
. . . . . . 7
⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
39 | 35, 38 | bitr3i 185 |
. . . . . 6
⊢
([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
40 | 34, 39 | bitrdi 195 |
. . . . 5
⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓)) |
41 | 33, 40 | imbi12d 233 |
. . . 4
⊢ (𝑧 = 𝑦 → ((𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 → [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐵 → 𝜓))) |
42 | 18, 19, 41 | cbval 1734 |
. . 3
⊢
(∀𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜓)) |
43 | 11, 42 | bitri 183 |
. 2
⊢
(∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜓)) |
44 | | df-ral 2440 |
. 2
⊢
(∀𝑥 ∈
𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
45 | | df-ral 2440 |
. 2
⊢
(∀𝑦 ∈
𝐵 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜓)) |
46 | 43, 44, 45 | 3bitr4i 211 |
1
⊢
(∀𝑥 ∈
𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓) |