| Step | Hyp | Ref
| Expression |
| 1 | | nfnf1 2154 |
. . . . . 6
⊢
Ⅎ𝑦Ⅎ𝑦𝜑 |
| 2 | 1 | nfal 2323 |
. . . . 5
⊢
Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑 |
| 3 | | equsb3 2103 |
. . . . . . 7
⊢ ([𝑣 / 𝑥]𝑥 = 𝑢 ↔ 𝑣 = 𝑢) |
| 4 | 3 | sblbis 2309 |
. . . . . 6
⊢ ([𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ([𝑣 / 𝑥]𝜑 ↔ 𝑣 = 𝑢)) |
| 5 | | wl-nfsbtv 37578 |
. . . . . . 7
⊢
(∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦[𝑣 / 𝑥]𝜑) |
| 6 | | nfvd 1915 |
. . . . . . 7
⊢
(∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦 𝑣 = 𝑢) |
| 7 | 5, 6 | nfbid 1902 |
. . . . . 6
⊢
(∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦([𝑣 / 𝑥]𝜑 ↔ 𝑣 = 𝑢)) |
| 8 | 4, 7 | nfxfrd 1854 |
. . . . 5
⊢
(∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦[𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢)) |
| 9 | | sbequ 2083 |
. . . . . 6
⊢ (𝑣 = 𝑦 → ([𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ [𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑢))) |
| 10 | 9 | a1i 11 |
. . . . 5
⊢
(∀𝑥Ⅎ𝑦𝜑 → (𝑣 = 𝑦 → ([𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ [𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑢)))) |
| 11 | 2, 8, 10 | cbvaldw 2340 |
. . . 4
⊢
(∀𝑥Ⅎ𝑦𝜑 → (∀𝑣[𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑢))) |
| 12 | | sb8v 2355 |
. . . . 5
⊢
(∀𝑥(𝜑 ↔ 𝑥 = 𝑢) ↔ ∀𝑣[𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢)) |
| 13 | 12 | bicomi 224 |
. . . 4
⊢
(∀𝑣[𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑢)) |
| 14 | | equsb3 2103 |
. . . . . 6
⊢ ([𝑦 / 𝑥]𝑥 = 𝑢 ↔ 𝑦 = 𝑢) |
| 15 | 14 | sblbis 2309 |
. . . . 5
⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢)) |
| 16 | 15 | albii 1819 |
. . . 4
⊢
(∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢)) |
| 17 | 11, 13, 16 | 3bitr3g 313 |
. . 3
⊢
(∀𝑥Ⅎ𝑦𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑢) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢))) |
| 18 | 17 | exbidv 1921 |
. 2
⊢
(∀𝑥Ⅎ𝑦𝜑 → (∃𝑢∀𝑥(𝜑 ↔ 𝑥 = 𝑢) ↔ ∃𝑢∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢))) |
| 19 | | eu6 2574 |
. 2
⊢
(∃!𝑥𝜑 ↔ ∃𝑢∀𝑥(𝜑 ↔ 𝑥 = 𝑢)) |
| 20 | | eu6 2574 |
. 2
⊢
(∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑢∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢)) |
| 21 | 18, 19, 20 | 3bitr4g 314 |
1
⊢
(∀𝑥Ⅎ𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)) |