Theorem sb8eu 1891

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 1398	. . . . 5 ⊢ Ⅎw(φ ↔ x = z)
2	1	sb8 1714	. . . 4 ⊢ (∀x(φ ↔ x = z) ↔ ∀w[w / x](φ ↔ x = z))
3		sbbi 1811	. . . . . 6 ⊢ ([w / x](φ ↔ x = z) ↔ ([w / x]φ ↔ [w / x]x = z))
4		sb8eu.1	. . . . . . . 8 ⊢ Ⅎyφ
5	4	nfsb 1800	. . . . . . 7 ⊢ Ⅎy[w / x]φ
6		equsb3 1803	. . . . . . . 8 ⊢ ([w / x]x = z ↔ w = z)
7		nfv 1398	. . . . . . . 8 ⊢ Ⅎy w = z
8	6, 7	nfxfr 1339	. . . . . . 7 ⊢ Ⅎy[w / x]x = z
9	5, 8	nfbi 1459	. . . . . 6 ⊢ Ⅎy([w / x]φ ↔ [w / x]x = z)
10	3, 9	nfxfr 1339	. . . . 5 ⊢ Ⅎy[w / x](φ ↔ x = z)
11		nfv 1398	. . . . 5 ⊢ Ⅎw[y / x](φ ↔ x = z)
12		sbequ 1699	. . . . 5 ⊢ (w = y → ([w / x](φ ↔ x = z) ↔ [y / x](φ ↔ x = z)))
13	10, 11, 12	cbval 1615	. . . 4 ⊢ (∀w[w / x](φ ↔ x = z) ↔ ∀y[y / x](φ ↔ x = z))
14		equsb3 1803	. . . . . 6 ⊢ ([y / x]x = z ↔ y = z)
15	14	sblbis 1812	. . . . 5 ⊢ ([y / x](φ ↔ x = z) ↔ ([y / x]φ ↔ y = z))
16	15	albii 1335	. . . 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 1474	. 2 ⊢ (∃z∀x(φ ↔ x = z) ↔ ∃z∀y([y / x]φ ↔ y = z))
19		df-eu 1881	. 2 ⊢ (∃!xφ ↔ ∃z∀x(φ ↔ x = z))
20		df-eu 1881	. 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 1224  Ⅎwnf 1325  ∃wex 1358  [wsb 1623  ∃!weu 1878

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406

This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881

This theorem is referenced by:  sb8mo  1892  nfeud  1894  nfeu  1897  cbveu  1902  cbvreu  2505  acexmid  5431