Theorem wl-sb8eut 38510

Description: Substitution of variable in universal quantifier. Closed form of sb8eu 2626. (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 2191	. . . . . 6 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑
2	1	nfal 2354	. . . . 5 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑
3		equsb3 2140	. . . . . . 7 ⊢ ([𝑣 / 𝑥]𝑥 = 𝑢 ↔ 𝑣 = 𝑢)
4	3	sblbis 2342	. . . . . 6 ⊢ ([𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ([𝑣 / 𝑥]𝜑 ↔ 𝑣 = 𝑢))
5		nfa1 2188	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑
6		sp 2220	. . . . . . . 8 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦𝜑)
7	5, 6	nfsbd 2552	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦[𝑣 / 𝑥]𝜑)
8		nfvd 1948	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦 𝑣 = 𝑢)
9	7, 8	nfbid 1935	. . . . . 6 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦([𝑣 / 𝑥]𝜑 ↔ 𝑣 = 𝑢))
10	4, 9	nfxfrd 1887	. . . . 5 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦[𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢))
11		sbequ 2120	. . . . . 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 2437	. . . 4 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑣[𝑣 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑢)))
14		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑣(𝜑 ↔ 𝑥 = 𝑢)
15	14	sb8 2547	. . . . 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 2140	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑢 ↔ 𝑦 = 𝑢)
18	17	sblbis 2342	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑢) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢))
19	18	albii 1852	. . . 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 1954	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑢∀𝑥(𝜑 ↔ 𝑥 = 𝑢) ↔ ∃𝑢∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑢)))
22		eu6 2600	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑢∀𝑥(𝜑 ↔ 𝑥 = 𝑢))
23		eu6 2600	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816  [wsb 2099  ∃!weu 2594

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-11 2194  ax-12 2213  ax-13 2402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595

This theorem is used by:  wl-sb8mot  38512