Theorem sb8eu 2503

Description: Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.)

Hypothesis

Ref	Expression
sb8eu.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
sb8eu	⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8eu

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . . . 5 ⊢ Ⅎ𝑤(𝜑 ↔ 𝑥 = 𝑧)
2	1	sb8 2424	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
3		equsb3 2432	. . . . . 6 ⊢ ([𝑤 / 𝑥]𝑥 = 𝑧 ↔ 𝑤 = 𝑧)
4	3	sblbis 2404	. . . . 5 ⊢ ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ 𝑤 = 𝑧))
5	4	albii 1747	. . . 4 ⊢ (∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑤([𝑤 / 𝑥]𝜑 ↔ 𝑤 = 𝑧))
6		sb8eu.1	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
7	6	nfsb 2440	. . . . . 6 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑
8		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑦 𝑤 = 𝑧
9	7, 8	nfbi 1833	. . . . 5 ⊢ Ⅎ𝑦([𝑤 / 𝑥]𝜑 ↔ 𝑤 = 𝑧)
10		nfv 1843	. . . . 5 ⊢ Ⅎ𝑤([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)
11		sbequ 2376	. . . . . 6 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
12		equequ1 1952	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝑧 ↔ 𝑦 = 𝑧))
13	11, 12	bibi12d 335	. . . . 5 ⊢ (𝑤 = 𝑦 → (([𝑤 / 𝑥]𝜑 ↔ 𝑤 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)))
14	9, 10, 13	cbval 2271	. . . 4 ⊢ (∀𝑤([𝑤 / 𝑥]𝜑 ↔ 𝑤 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
15	2, 5, 14	3bitri 286	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
16	15	exbii 1774	. 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
17		df-eu 2474	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
18		df-eu 2474	. 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑧∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
19	16, 17, 18	3bitr4i 292	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 196  ∀wal 1481  ∃wex 1704  Ⅎwnf 1708  [wsb 1880  ∃!weu 2470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474

This theorem is referenced by:  sb8mo  2504  cbveu  2505  eu1  2510  cbvreu  3169