Theorem wl-sb8eut 38121

Description: Substitution of variable in universal quantifier. Closed form of sb8eu 2634. (Contributed by Wolf Lammen, 11-Aug-2019.)

Assertion

Ref	Expression
wl-sb8eut	⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))

Proof of Theorem wl-sb8eut

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfnf1 2195	. . . . . 6 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑
2	1	nfal 2362	. . . . 5 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑
3		equsb3 2144	. . . . . . 7 ⊢ ([𝑣 / 𝑥]𝑥 = 𝑢 ↔ 𝑣 = 𝑢)
4	3	sblbis 2349	. . . . . 6 ⊢ ([𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ([𝑣 / 𝑥]𝜑 ↔ 𝑣 = 𝑢))
5		nfa1 2192	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑
6		sp 2225	. . . . . . . 8 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦𝜑)
7	5, 6	nfsbd 2560	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦[𝑣 / 𝑥]𝜑)
8		nfvd 1942	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦 𝑣 = 𝑢)
9	7, 8	nfbid 1929	. . . . . 6 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦([𝑣 / 𝑥]𝜑 ↔ 𝑣 = 𝑢))
10	4, 9	nfxfrd 1881	. . . . 5 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦[𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢))
11		sbequ 2123	. . . . . 6 ⊢ (𝑣 = 𝑦 → ([𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ [𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑢)))
12	11	a1i 11	. . . . 5 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (𝑣 = 𝑦 → ([𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ [𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑢))))
13	2, 10, 12	cbvald 2445	. . . 4 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑣[𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑢)))
14		nfv 1941	. . . . . 6 ⊢ Ⅎ𝑣(𝜑 ↔ 𝑥 = 𝑢)
15	14	sb8 2555	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑢) ↔ ∀𝑣[𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢))
16	15	bicomi 227	. . . 4 ⊢ (∀𝑣[𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑢))
17		equsb3 2144	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑢 ↔ 𝑦 = 𝑢)
18	17	sblbis 2349	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢))
19	18	albii 1846	. . . 4 ⊢ (∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢))
20	13, 16, 19	3bitr3g 316	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑢) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢)))
21	20	exbidv 1948	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑢∀𝑥(𝜑 ↔ 𝑥 = 𝑢) ↔ ∃𝑢∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢)))
22		eu6 2608	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑢∀𝑥(𝜑 ↔ 𝑥 = 𝑢))
23		eu6 2608	. 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑢∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢))
24	21, 22, 23	3bitr4g 317	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565  ∃wex 1806  Ⅎwnf 1810  [wsb 2097  ∃!weu 2602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-11 2198  ax-12 2219  ax-13 2410

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603

This theorem is referenced by:  wl-sb8mot  38123