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Theorem wl-sb8eut 33330
Description: Substitution of variable in universal quantifier. Closed form of sb8eu 2501. (Contributed by Wolf Lammen, 11-Aug-2019.)
Assertion
Ref Expression
wl-sb8eut (∀𝑥𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))

Proof of Theorem wl-sb8eut
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfnf1 2029 . . . . . 6 𝑦𝑦𝜑
21nfal 2151 . . . . 5 𝑦𝑥𝑦𝜑
3 equsb3 2430 . . . . . . 7 ([𝑣 / 𝑥]𝑥 = 𝑢𝑣 = 𝑢)
43sblbis 2402 . . . . . 6 ([𝑣 / 𝑥](𝜑𝑥 = 𝑢) ↔ ([𝑣 / 𝑥]𝜑𝑣 = 𝑢))
5 nfa1 2026 . . . . . . . 8 𝑥𝑥𝑦𝜑
6 sp 2051 . . . . . . . 8 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝜑)
75, 6nfsbd 2440 . . . . . . 7 (∀𝑥𝑦𝜑 → Ⅎ𝑦[𝑣 / 𝑥]𝜑)
8 nfvd 1842 . . . . . . 7 (∀𝑥𝑦𝜑 → Ⅎ𝑦 𝑣 = 𝑢)
97, 8nfbid 1830 . . . . . 6 (∀𝑥𝑦𝜑 → Ⅎ𝑦([𝑣 / 𝑥]𝜑𝑣 = 𝑢))
104, 9nfxfrd 1778 . . . . 5 (∀𝑥𝑦𝜑 → Ⅎ𝑦[𝑣 / 𝑥](𝜑𝑥 = 𝑢))
11 sbequ 2374 . . . . . 6 (𝑣 = 𝑦 → ([𝑣 / 𝑥](𝜑𝑥 = 𝑢) ↔ [𝑦 / 𝑥](𝜑𝑥 = 𝑢)))
1211a1i 11 . . . . 5 (∀𝑥𝑦𝜑 → (𝑣 = 𝑦 → ([𝑣 / 𝑥](𝜑𝑥 = 𝑢) ↔ [𝑦 / 𝑥](𝜑𝑥 = 𝑢))))
132, 10, 12cbvald 2275 . . . 4 (∀𝑥𝑦𝜑 → (∀𝑣[𝑣 / 𝑥](𝜑𝑥 = 𝑢) ↔ ∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑢)))
14 nfv 1841 . . . . . 6 𝑣(𝜑𝑥 = 𝑢)
1514sb8 2422 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑢) ↔ ∀𝑣[𝑣 / 𝑥](𝜑𝑥 = 𝑢))
1615bicomi 214 . . . 4 (∀𝑣[𝑣 / 𝑥](𝜑𝑥 = 𝑢) ↔ ∀𝑥(𝜑𝑥 = 𝑢))
17 equsb3 2430 . . . . . 6 ([𝑦 / 𝑥]𝑥 = 𝑢𝑦 = 𝑢)
1817sblbis 2402 . . . . 5 ([𝑦 / 𝑥](𝜑𝑥 = 𝑢) ↔ ([𝑦 / 𝑥]𝜑𝑦 = 𝑢))
1918albii 1745 . . . 4 (∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑢) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑢))
2013, 16, 193bitr3g 302 . . 3 (∀𝑥𝑦𝜑 → (∀𝑥(𝜑𝑥 = 𝑢) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑢)))
2120exbidv 1848 . 2 (∀𝑥𝑦𝜑 → (∃𝑢𝑥(𝜑𝑥 = 𝑢) ↔ ∃𝑢𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑢)))
22 df-eu 2472 . 2 (∃!𝑥𝜑 ↔ ∃𝑢𝑥(𝜑𝑥 = 𝑢))
23 df-eu 2472 . 2 (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑢𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑢))
2421, 22, 233bitr4g 303 1 (∀𝑥𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1479  wex 1702  wnf 1706  [wsb 1878  ∃!weu 2468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472
This theorem is referenced by:  wl-sb8mot  33331
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