Theorem sb8eu 2222

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 1619	. . . . 5 ⊢ Ⅎw(φ ↔ x = z)
2	1	sb8 2092	. . . 4 ⊢ (∀x(φ ↔ x = z) ↔ ∀w[w / x](φ ↔ x = z))
3		sbbi 2071	. . . . . 6 ⊢ ([w / x](φ ↔ x = z) ↔ ([w / x]φ ↔ [w / x]x = z))
4		sb8eu.1	. . . . . . . 8 ⊢ Ⅎyφ
5	4	nfsb 2109	. . . . . . 7 ⊢ Ⅎy[w / x]φ
6		equsb3 2102	. . . . . . . 8 ⊢ ([w / x]x = z ↔ w = z)
7		nfv 1619	. . . . . . . 8 ⊢ Ⅎy w = z
8	6, 7	nfxfr 1570	. . . . . . 7 ⊢ Ⅎy[w / x]x = z
9	5, 8	nfbi 1834	. . . . . 6 ⊢ Ⅎy([w / x]φ ↔ [w / x]x = z)
10	3, 9	nfxfr 1570	. . . . 5 ⊢ Ⅎy[w / x](φ ↔ x = z)
11		nfv 1619	. . . . 5 ⊢ Ⅎw[y / x](φ ↔ x = z)
12		sbequ 2060	. . . . 5 ⊢ (w = y → ([w / x](φ ↔ x = z) ↔ [y / x](φ ↔ x = z)))
13	10, 11, 12	cbval 1984	. . . 4 ⊢ (∀w[w / x](φ ↔ x = z) ↔ ∀y[y / x](φ ↔ x = z))
14		equsb3 2102	. . . . . 6 ⊢ ([y / x]x = z ↔ y = z)
15	14	sblbis 2072	. . . . 5 ⊢ ([y / x](φ ↔ x = z) ↔ ([y / x]φ ↔ y = z))
16	15	albii 1566	. . . 4 ⊢ (∀y[y / x](φ ↔ x = z) ↔ ∀y([y / x]φ ↔ y = z))
17	2, 13, 16	3bitri 262	. . 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 2208	. 2 ⊢ (∃!xφ ↔ ∃z∀x(φ ↔ x = z))
20		df-eu 2208	. 2 ⊢ (∃!y[y / x]φ ↔ ∃z∀y([y / x]φ ↔ y = z))
21	18, 19, 20	3bitr4i 268	1 ⊢ (∃!xφ ↔ ∃!y[y / x]φ)

Syntax hints:   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  [wsb 1648  ∃!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208

This theorem is referenced by:  sb8mo  2223  cbveu  2224  eu1  2225  cbvreu  2833