Theorem wl-sb8eutv 37533

Description: Substitution of variable in universal quantifier. Closed form of sb8euv 2602. (Contributed by Wolf Lammen, 3-May-2025.)

Assertion

Ref	Expression
wl-sb8eutv	⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃!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.

Step	Hyp	Ref	Expression
1		nfnf1 2155	. . . . . 6 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑
2	1	nfal 2327	. . . . 5 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑
3		equsb3 2103	. . . . . . 7 ⊢ ([𝑣 / 𝑥]𝑥 = 𝑢 ↔ 𝑣 = 𝑢)
4	3	sblbis 2313	. . . . . 6 ⊢ ([𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ([𝑣 / 𝑥]𝜑 ↔ 𝑣 = 𝑢))
5		wl-nfsbtv 37531	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦[𝑣 / 𝑥]𝜑)
6		nfvd 1914	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦 𝑣 = 𝑢)
7	5, 6	nfbid 1901	. . . . . 6 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦([𝑣 / 𝑥]𝜑 ↔ 𝑣 = 𝑢))
8	4, 7	nfxfrd 1852	. . . . 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 2344	. . . 4 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑣[𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑢)))
12		sb8v 2358	. . . . 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 2313	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢))
16	15	albii 1817	. . . 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 1920	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑢∀𝑥(𝜑 ↔ 𝑥 = 𝑢) ↔ ∃𝑢∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢)))
19		eu6 2577	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑢∀𝑥(𝜑 ↔ 𝑥 = 𝑢))
20		eu6 2577	. 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))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535  ∃wex 1777  Ⅎwnf 1781  [wsb 2064  ∃!weu 2571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572

This theorem is referenced by:  wl-sb8motv  37535