Theorem bm1.1 2338

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 1619	. . . . . . . 8 ⊢ Ⅎx y ∈ z
2		bm1.1.1	. . . . . . . 8 ⊢ Ⅎxφ
3	1, 2	nfbi 1834	. . . . . . 7 ⊢ Ⅎx(y ∈ z ↔ φ)
4	3	nfal 1842	. . . . . 6 ⊢ Ⅎx∀y(y ∈ z ↔ φ)
5		elequ2 1715	. . . . . . . 8 ⊢ (x = z → (y ∈ x ↔ y ∈ z))
6	5	bibi1d 310	. . . . . . 7 ⊢ (x = z → ((y ∈ x ↔ φ) ↔ (y ∈ z ↔ φ)))
7	6	albidv 1625	. . . . . 6 ⊢ (x = z → (∀y(y ∈ x ↔ φ) ↔ ∀y(y ∈ z ↔ φ)))
8	4, 7	sbie 2038	. . . . 5 ⊢ ([z / x]∀y(y ∈ x ↔ φ) ↔ ∀y(y ∈ z ↔ φ))
9		19.26 1593	. . . . . 6 ⊢ (∀y((y ∈ x ↔ φ) ∧ (y ∈ z ↔ φ)) ↔ (∀y(y ∈ x ↔ φ) ∧ ∀y(y ∈ z ↔ φ)))
10		biantr 897	. . . . . . . 8 ⊢ (((y ∈ x ↔ φ) ∧ (y ∈ z ↔ φ)) → (y ∈ x ↔ y ∈ z))
11	10	alimi 1559	. . . . . . 7 ⊢ (∀y((y ∈ x ↔ φ) ∧ (y ∈ z ↔ φ)) → ∀y(y ∈ x ↔ y ∈ z))
12		ax-ext 2334	. . . . . . 7 ⊢ (∀y(y ∈ x ↔ y ∈ z) → x = z)
13	11, 12	syl 15	. . . . . 6 ⊢ (∀y((y ∈ x ↔ φ) ∧ (y ∈ z ↔ φ)) → x = z)
14	9, 13	sylbir 204	. . . . 5 ⊢ ((∀y(y ∈ x ↔ φ) ∧ ∀y(y ∈ z ↔ φ)) → x = z)
15	8, 14	sylan2b 461	. . . 4 ⊢ ((∀y(y ∈ x ↔ φ) ∧ [z / x]∀y(y ∈ x ↔ φ)) → x = z)
16	15	gen2 1547	. . 3 ⊢ ∀x∀z((∀y(y ∈ x ↔ φ) ∧ [z / x]∀y(y ∈ x ↔ φ)) → x = z)
17	16	jctr 526	. 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 1619	. . 3 ⊢ Ⅎz∀y(y ∈ x ↔ φ)
19	18	eu2 2229	. 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 203	1 ⊢ (∃x∀y(y ∈ x ↔ φ) → ∃!x∀y(y ∈ x ↔ φ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  ∃!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-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

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: (None)