Theorem sb8eu 1910

Description: Variable substitution in uniqueness 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 1418	. . . . 5 ⊢ Ⅎw(φ ↔ x = z)
2	1	sb8 1733	. . . 4 ⊢ (∀x(φ ↔ x = z) ↔ ∀w[w / x](φ ↔ x = z))
3		sbbi 1830	. . . . . 6 ⊢ ([w / x](φ ↔ x = z) ↔ ([w / x]φ ↔ [w / x]x = z))
4		sb8eu.1	. . . . . . . 8 ⊢ Ⅎyφ
5	4	nfsb 1819	. . . . . . 7 ⊢ Ⅎy[w / x]φ
6		equsb3 1822	. . . . . . . 8 ⊢ ([w / x]x = z ↔ w = z)
7		nfv 1418	. . . . . . . 8 ⊢ Ⅎy w = z
8	6, 7	nfxfr 1360	. . . . . . 7 ⊢ Ⅎy[w / x]x = z
9	5, 8	nfbi 1478	. . . . . 6 ⊢ Ⅎy([w / x]φ ↔ [w / x]x = z)
10	3, 9	nfxfr 1360	. . . . 5 ⊢ Ⅎy[w / x](φ ↔ x = z)
11		nfv 1418	. . . . 5 ⊢ Ⅎw[y / x](φ ↔ x = z)
12		sbequ 1718	. . . . 5 ⊢ (w = y → ([w / x](φ ↔ x = z) ↔ [y / x](φ ↔ x = z)))
13	10, 11, 12	cbval 1634	. . . 4 ⊢ (∀w[w / x](φ ↔ x = z) ↔ ∀y[y / x](φ ↔ x = z))
14		equsb3 1822	. . . . . 6 ⊢ ([y / x]x = z ↔ y = z)
15	14	sblbis 1831	. . . . 5 ⊢ ([y / x](φ ↔ x = z) ↔ ([y / x]φ ↔ y = z))
16	15	albii 1356	. . . 4 ⊢ (∀y[y / x](φ ↔ x = z) ↔ ∀y([y / x]φ ↔ y = z))
17	2, 13, 16	3bitri 195	. . 3 ⊢ (∀x(φ ↔ x = z) ↔ ∀y([y / x]φ ↔ y = z))
18	17	exbii 1493	. 2 ⊢ (∃z∀x(φ ↔ x = z) ↔ ∃z∀y([y / x]φ ↔ y = z))
19		df-eu 1900	. 2 ⊢ (∃!xφ ↔ ∃z∀x(φ ↔ x = z))
20		df-eu 1900	. 2 ⊢ (∃!y[y / x]φ ↔ ∃z∀y([y / x]φ ↔ y = z))
21	18, 19, 20	3bitr4i 201	1 ⊢ (∃!xφ ↔ ∃!y[y / x]φ)

Syntax hints:   ↔ wb 98  ∀wal 1240  Ⅎwnf 1346  ∃wex 1378  [wsb 1642  ∃!weu 1897

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900

This theorem is referenced by:  sb8mo  1911  nfeud  1913  nfeu  1916  cbveu  1921  cbvreu  2525  acexmid  5454