Theorem bm1.1 2011

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 1408	. . . . . . . 8 ⊢ Ⅎx y ∈ z
2		bm1.1.1	. . . . . . . 8 ⊢ Ⅎxφ
3	1, 2	nfbi 1470	. . . . . . 7 ⊢ Ⅎx(y ∈ z ↔ φ)
4	3	nfal 1457	. . . . . 6 ⊢ Ⅎx∀y(y ∈ z ↔ φ)
5		elequ2 1589	. . . . . . . 8 ⊢ (x = z → (y ∈ x ↔ y ∈ z))
6	5	bibi1d 222	. . . . . . 7 ⊢ (x = z → ((y ∈ x ↔ φ) ↔ (y ∈ z ↔ φ)))
7	6	albidv 1693	. . . . . 6 ⊢ (x = z → (∀y(y ∈ x ↔ φ) ↔ ∀y(y ∈ z ↔ φ)))
8	4, 7	sbie 1662	. . . . 5 ⊢ ([z / x]∀y(y ∈ x ↔ φ) ↔ ∀y(y ∈ z ↔ φ))
9		19.26 1355	. . . . . 6 ⊢ (∀y((y ∈ x ↔ φ) ∧ (y ∈ z ↔ φ)) ↔ (∀y(y ∈ x ↔ φ) ∧ ∀y(y ∈ z ↔ φ)))
10		biantr 852	. . . . . . . 8 ⊢ (((y ∈ x ↔ φ) ∧ (y ∈ z ↔ φ)) → (y ∈ x ↔ y ∈ z))
11	10	alimi 1329	. . . . . . 7 ⊢ (∀y((y ∈ x ↔ φ) ∧ (y ∈ z ↔ φ)) → ∀y(y ∈ x ↔ y ∈ z))
12		ax-ext 2008	. . . . . . 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 1324	. . 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 1408	. . 3 ⊢ Ⅎz∀y(y ∈ x ↔ φ)
19	18	eu2 1930	. 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 1231  Ⅎwnf 1334  ∃wex 1368  [wsb 1633  ∃!weu 1886

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 1321  ax-7 1322  ax-gen 1323  ax-ie1 1369  ax-ie2 1370  ax-8 1382  ax-10 1383  ax-11 1384  ax-i12 1385  ax-bnd 1386  ax-4 1387  ax-14 1392  ax-17 1406  ax-i9 1410  ax-ial 1415  ax-i5r 1416  ax-ext 2008

This theorem depends on definitions:  df-bi 110  df-tru 1236  df-nf 1335  df-sb 1634  df-eu 1889

This theorem is referenced by:  zfnuleu  3857