Theorem sb8eu 2058

Description: Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

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 1542	. . . . 5 ⊢ Ⅎ𝑤(𝜑 ↔ 𝑥 = 𝑧)
2	1	sb8 1870	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
3		sbbi 1978	. . . . . 6 ⊢ ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
4		sb8eu.1	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
5	4	nfsb 1965	. . . . . . 7 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑
6		equsb3 1970	. . . . . . . 8 ⊢ ([𝑤 / 𝑥]𝑥 = 𝑧 ↔ 𝑤 = 𝑧)
7		nfv 1542	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑤 = 𝑧
8	6, 7	nfxfr 1488	. . . . . . 7 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝑥 = 𝑧
9	5, 8	nfbi 1603	. . . . . 6 ⊢ Ⅎ𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧)
10	3, 9	nfxfr 1488	. . . . 5 ⊢ Ⅎ𝑦[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)
11		nfv 1542	. . . . 5 ⊢ Ⅎ𝑤[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)
12		sbequ 1854	. . . . 5 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ [𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)))
13	10, 11, 12	cbval 1768	. . . 4 ⊢ (∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
14		equsb3 1970	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)
15	14	sblbis 1979	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
16	15	albii 1484	. . . 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 1619	. 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
19		df-eu 2048	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
20		df-eu 2048	. 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:   ↔ wb 105  ∀wal 1362  Ⅎwnf 1474  ∃wex 1506  [wsb 1776  ∃!weu 2045

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048

This theorem is referenced by:  sb8mo  2059  nfeud  2061  nfeu  2064  cbveu  2069  cbvreu  2727  acexmid  5921