Theorem bm1.1 2007

Description: Any set defined by a property is the only set defined by that property. Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM, 30-Jun-1994.)

Hypothesis

Ref	Expression
bm1.1.1	⊢ Ⅎxφ

Assertion

Ref	Expression
bm1.1	⊢ (∃x∀y(y ∈ x ↔ φ) → ∃!x∀y(y ∈ x ↔ φ))

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem bm1.1

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1402	. . . . . . . 8 ⊢ Ⅎx y ∈ z
2		bm1.1.1	. . . . . . . 8 ⊢ Ⅎxφ
3	1, 2	nfbi 1463	. . . . . . 7 ⊢ Ⅎx(y ∈ z ↔ φ)
4	3	nfal 1450	. . . . . 6 ⊢ Ⅎx∀y(y ∈ z ↔ φ)
5		elequ2 1583	. . . . . . . 8 ⊢ (x = z → (y ∈ x ↔ y ∈ z))
6	5	bibi1d 222	. . . . . . 7 ⊢ (x = z → ((y ∈ x ↔ φ) ↔ (y ∈ z ↔ φ)))
7	6	albidv 1687	. . . . . 6 ⊢ (x = z → (∀y(y ∈ x ↔ φ) ↔ ∀y(y ∈ z ↔ φ)))
8	4, 7	sbie 1656	. . . . 5 ⊢ ([z / x]∀y(y ∈ x ↔ φ) ↔ ∀y(y ∈ z ↔ φ))
9		19.26 1350	. . . . . 6 ⊢ (∀y((y ∈ x ↔ φ) ∧ (y ∈ z ↔ φ)) ↔ (∀y(y ∈ x ↔ φ) ∧ ∀y(y ∈ z ↔ φ)))
10		biantr 847	. . . . . . . 8 ⊢ (((y ∈ x ↔ φ) ∧ (y ∈ z ↔ φ)) → (y ∈ x ↔ y ∈ z))
11	10	alimi 1324	. . . . . . 7 ⊢ (∀y((y ∈ x ↔ φ) ∧ (y ∈ z ↔ φ)) → ∀y(y ∈ x ↔ y ∈ z))
12		ax-ext 2004	. . . . . . 7 ⊢ (∀y(y ∈ x ↔ y ∈ z) → x = z)
13	11, 12	syl 14	. . . . . 6 ⊢ (∀y((y ∈ x ↔ φ) ∧ (y ∈ z ↔ φ)) → x = z)
14	9, 13	sylbir 125	. . . . 5 ⊢ ((∀y(y ∈ x ↔ φ) ∧ ∀y(y ∈ z ↔ φ)) → x = z)
15	8, 14	sylan2b 271	. . . 4 ⊢ ((∀y(y ∈ x ↔ φ) ∧ [z / x]∀y(y ∈ x ↔ φ)) → x = z)
16	15	gen2 1319	. . 3 ⊢ ∀x∀z((∀y(y ∈ x ↔ φ) ∧ [z / x]∀y(y ∈ x ↔ φ)) → x = z)
17	16	jctr 298	. 2 ⊢ (∃x∀y(y ∈ x ↔ φ) → (∃x∀y(y ∈ x ↔ φ) ∧ ∀x∀z((∀y(y ∈ x ↔ φ) ∧ [z / x]∀y(y ∈ x ↔ φ)) → x = z)))
18		nfv 1402	. . 3 ⊢ Ⅎz∀y(y ∈ x ↔ φ)
19	18	eu2 1926	. 2 ⊢ (∃!x∀y(y ∈ x ↔ φ) ↔ (∃x∀y(y ∈ x ↔ φ) ∧ ∀x∀z((∀y(y ∈ x ↔ φ) ∧ [z / x]∀y(y ∈ x ↔ φ)) → x = z)))
20	17, 19	sylibr 137	1 ⊢ (∃x∀y(y ∈ x ↔ φ) → ∃!x∀y(y ∈ x ↔ φ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1226  Ⅎwnf 1329  ∃wex 1362  [wsb 1627  ∃!weu 1882

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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885

This theorem is referenced by:  zfnuleu  3855