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Theorem wl-sb8eutv 37559
Description: Substitution of variable in universal quantifier. Closed form of sb8euv 2596. (Contributed by Wolf Lammen, 3-May-2025.)
Assertion
Ref Expression
wl-sb8eutv (∀𝑥𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem wl-sb8eutv
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfnf1 2151 . . . . . 6 𝑦𝑦𝜑
21nfal 2321 . . . . 5 𝑦𝑥𝑦𝜑
3 equsb3 2100 . . . . . . 7 ([𝑣 / 𝑥]𝑥 = 𝑢𝑣 = 𝑢)
43sblbis 2307 . . . . . 6 ([𝑣 / 𝑥](𝜑𝑥 = 𝑢) ↔ ([𝑣 / 𝑥]𝜑𝑣 = 𝑢))
5 wl-nfsbtv 37557 . . . . . . 7 (∀𝑥𝑦𝜑 → Ⅎ𝑦[𝑣 / 𝑥]𝜑)
6 nfvd 1912 . . . . . . 7 (∀𝑥𝑦𝜑 → Ⅎ𝑦 𝑣 = 𝑢)
75, 6nfbid 1899 . . . . . 6 (∀𝑥𝑦𝜑 → Ⅎ𝑦([𝑣 / 𝑥]𝜑𝑣 = 𝑢))
84, 7nfxfrd 1850 . . . . 5 (∀𝑥𝑦𝜑 → Ⅎ𝑦[𝑣 / 𝑥](𝜑𝑥 = 𝑢))
9 sbequ 2080 . . . . . 6 (𝑣 = 𝑦 → ([𝑣 / 𝑥](𝜑𝑥 = 𝑢) ↔ [𝑦 / 𝑥](𝜑𝑥 = 𝑢)))
109a1i 11 . . . . 5 (∀𝑥𝑦𝜑 → (𝑣 = 𝑦 → ([𝑣 / 𝑥](𝜑𝑥 = 𝑢) ↔ [𝑦 / 𝑥](𝜑𝑥 = 𝑢))))
112, 8, 10cbvaldw 2338 . . . 4 (∀𝑥𝑦𝜑 → (∀𝑣[𝑣 / 𝑥](𝜑𝑥 = 𝑢) ↔ ∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑢)))
12 sb8v 2352 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑢) ↔ ∀𝑣[𝑣 / 𝑥](𝜑𝑥 = 𝑢))
1312bicomi 224 . . . 4 (∀𝑣[𝑣 / 𝑥](𝜑𝑥 = 𝑢) ↔ ∀𝑥(𝜑𝑥 = 𝑢))
14 equsb3 2100 . . . . . 6 ([𝑦 / 𝑥]𝑥 = 𝑢𝑦 = 𝑢)
1514sblbis 2307 . . . . 5 ([𝑦 / 𝑥](𝜑𝑥 = 𝑢) ↔ ([𝑦 / 𝑥]𝜑𝑦 = 𝑢))
1615albii 1815 . . . 4 (∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑢) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑢))
1711, 13, 163bitr3g 313 . . 3 (∀𝑥𝑦𝜑 → (∀𝑥(𝜑𝑥 = 𝑢) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑢)))
1817exbidv 1918 . 2 (∀𝑥𝑦𝜑 → (∃𝑢𝑥(𝜑𝑥 = 𝑢) ↔ ∃𝑢𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑢)))
19 eu6 2571 . 2 (∃!𝑥𝜑 ↔ ∃𝑢𝑥(𝜑𝑥 = 𝑢))
20 eu6 2571 . 2 (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑢𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑢))
2118, 19, 203bitr4g 314 1 (∀𝑥𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1534  wex 1775  wnf 1779  [wsb 2061  ∃!weu 2565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-10 2138  ax-11 2154  ax-12 2174
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566
This theorem is referenced by:  wl-sb8motv  37561
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