Theorem wl-sb8eut 35659

Description: Substitution of variable in universal quantifier. Closed form of sb8eu 2600. (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 2153	. . . . . 6 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑
2	1	nfal 2321	. . . . 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		nfa1 2150	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑
6		sp 2178	. . . . . . . 8 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦𝜑)
7	5, 6	nfsbd 2526	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦[𝑣 / 𝑥]𝜑)
8		nfvd 1919	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦 𝑣 = 𝑢)
9	7, 8	nfbid 1906	. . . . . 6 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦([𝑣 / 𝑥]𝜑 ↔ 𝑣 = 𝑢))
10	4, 9	nfxfrd 1857	. . . . 5 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦[𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢))
11		sbequ 2087	. . . . . 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 2407	. . . 4 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑣[𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑢)))
14		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑣(𝜑 ↔ 𝑥 = 𝑢)
15	14	sb8 2521	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑢) ↔ ∀𝑣[𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢))
16	15	bicomi 223	. . . 4 ⊢ (∀𝑣[𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑢))
17		equsb3 2103	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑢 ↔ 𝑦 = 𝑢)
18	17	sblbis 2309	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢))
19	18	albii 1823	. . . 4 ⊢ (∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢))
20	13, 16, 19	3bitr3g 312	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑢) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢)))
21	20	exbidv 1925	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑢∀𝑥(𝜑 ↔ 𝑥 = 𝑢) ↔ ∃𝑢∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢)))
22		eu6 2574	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑢∀𝑥(𝜑 ↔ 𝑥 = 𝑢))
23		eu6 2574	. 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑢∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢))
24	21, 22, 23	3bitr4g 313	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1783  Ⅎwnf 1787  [wsb 2068  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569

This theorem is referenced by:  wl-sb8mot  35660