Theorem sb8eu 2016

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 1505	. . . . 5 ⊢ Ⅎ𝑤(𝜑 ↔ 𝑥 = 𝑧)
2	1	sb8 1833	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
3		sbbi 1936	. . . . . 6 ⊢ ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
4		sb8eu.1	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
5	4	nfsb 1923	. . . . . . 7 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑
6		equsb3 1928	. . . . . . . 8 ⊢ ([𝑤 / 𝑥]𝑥 = 𝑧 ↔ 𝑤 = 𝑧)
7		nfv 1505	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑤 = 𝑧
8	6, 7	nfxfr 1451	. . . . . . 7 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝑥 = 𝑧
9	5, 8	nfbi 1566	. . . . . 6 ⊢ Ⅎ𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧)
10	3, 9	nfxfr 1451	. . . . 5 ⊢ Ⅎ𝑦[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)
11		nfv 1505	. . . . 5 ⊢ Ⅎ𝑤[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)
12		sbequ 1817	. . . . 5 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ [𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)))
13	10, 11, 12	cbval 1731	. . . 4 ⊢ (∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
14		equsb3 1928	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)
15	14	sblbis 1937	. . . . 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 1582	. 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
19		df-eu 2006	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
20		df-eu 2006	. 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:   ↔ wb 104  ∀wal 1330  Ⅎwnf 1437  ∃wex 1469  [wsb 1739  ∃!weu 2003

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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006

This theorem is referenced by:  sb8mo  2017  nfeud  2019  nfeu  2022  cbveu  2027  cbvreu  2675  acexmid  5813