Theorem sb8euh 2049

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 1526	. . . . 5 ⊢ ((𝜑 ↔ 𝑥 = 𝑧) → ∀𝑤(𝜑 ↔ 𝑥 = 𝑧))
2	1	sb8h 1854	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
3		sbbi 1959	. . . . . 6 ⊢ ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
4		sb8euh.1	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦𝜑)
5	4	hbsb 1949	. . . . . . 7 ⊢ ([𝑤 / 𝑥]𝜑 → ∀𝑦[𝑤 / 𝑥]𝜑)
6		equsb3 1951	. . . . . . . 8 ⊢ ([𝑤 / 𝑥]𝑥 = 𝑧 ↔ 𝑤 = 𝑧)
7		ax-17 1526	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → ∀𝑦 𝑤 = 𝑧)
8	6, 7	hbxfrbi 1472	. . . . . . 7 ⊢ ([𝑤 / 𝑥]𝑥 = 𝑧 → ∀𝑦[𝑤 / 𝑥]𝑥 = 𝑧)
9	5, 8	hbbi 1548	. . . . . 6 ⊢ (([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧) → ∀𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
10	3, 9	hbxfrbi 1472	. . . . 5 ⊢ ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) → ∀𝑦[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
11		ax-17 1526	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) → ∀𝑤[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
12		sbequ 1840	. . . . 5 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ [𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)))
13	10, 11, 12	cbvalh 1753	. . . 4 ⊢ (∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
14		equsb3 1951	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)
15	14	sblbis 1960	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
16	15	albii 1470	. . . 4 ⊢ (∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
17	2, 13, 16	3bitri 206	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
18	17	exbii 1605	. 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
19		df-eu 2029	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
20		df-eu 2029	. 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑧∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
21	18, 19, 20	3bitr4i 212	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1351  ∃wex 1492  [wsb 1762  ∃!weu 2026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029

This theorem is referenced by:  eu1  2051