Theorem sb8euh 2023

Description: Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.)

Hypothesis

Ref	Expression
sb8euh.1	⊢ (𝜑 → ∀𝑦𝜑)

Assertion

Ref	Expression
sb8euh	⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8euh

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

Step	Hyp	Ref	Expression
1		ax-17 1507	. . . . 5 ⊢ ((𝜑 ↔ 𝑥 = 𝑧) → ∀𝑤(𝜑 ↔ 𝑥 = 𝑧))
2	1	sb8h 1827	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
3		sbbi 1933	. . . . . 6 ⊢ ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
4		sb8euh.1	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦𝜑)
5	4	hbsb 1923	. . . . . . 7 ⊢ ([𝑤 / 𝑥]𝜑 → ∀𝑦[𝑤 / 𝑥]𝜑)
6		equsb3 1925	. . . . . . . 8 ⊢ ([𝑤 / 𝑥]𝑥 = 𝑧 ↔ 𝑤 = 𝑧)
7		ax-17 1507	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → ∀𝑦 𝑤 = 𝑧)
8	6, 7	hbxfrbi 1449	. . . . . . 7 ⊢ ([𝑤 / 𝑥]𝑥 = 𝑧 → ∀𝑦[𝑤 / 𝑥]𝑥 = 𝑧)
9	5, 8	hbbi 1528	. . . . . 6 ⊢ (([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧) → ∀𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
10	3, 9	hbxfrbi 1449	. . . . 5 ⊢ ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) → ∀𝑦[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
11		ax-17 1507	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) → ∀𝑤[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
12		sbequ 1813	. . . . 5 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ [𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)))
13	10, 11, 12	cbvalh 1727	. . . 4 ⊢ (∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
14		equsb3 1925	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)
15	14	sblbis 1934	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
16	15	albii 1447	. . . 4 ⊢ (∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
17	2, 13, 16	3bitri 205	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
18	17	exbii 1585	. 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
19		df-eu 2003	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
20		df-eu 2003	. 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑧∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
21	18, 19, 20	3bitr4i 211	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1330  ∃wex 1469  [wsb 1736  ∃!weu 2000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003

This theorem is referenced by:  eu1  2025