Theorem bm1.1 2125

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 1509	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
2		bm1.1.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
3	1, 2	nfbi 1569	. . . . . . 7 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑧 ↔ 𝜑)
4	3	nfal 1556	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)
5		elequ2 1692	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
6	5	bibi1d 232	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑦 ∈ 𝑥 ↔ 𝜑) ↔ (𝑦 ∈ 𝑧 ↔ 𝜑)))
7	6	albidv 1797	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)))
8	4, 7	sbie 1765	. . . . 5 ⊢ ([𝑧 / 𝑥]∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑))
9		19.26 1458	. . . . . 6 ⊢ (∀𝑦((𝑦 ∈ 𝑥 ↔ 𝜑) ∧ (𝑦 ∈ 𝑧 ↔ 𝜑)) ↔ (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)))
10		biantr 937	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝑥 ↔ 𝜑) ∧ (𝑦 ∈ 𝑧 ↔ 𝜑)) → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
11	10	alimi 1432	. . . . . . 7 ⊢ (∀𝑦((𝑦 ∈ 𝑥 ↔ 𝜑) ∧ (𝑦 ∈ 𝑧 ↔ 𝜑)) → ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
12		ax-ext 2122	. . . . . . 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 134	. . . . 5 ⊢ ((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → 𝑥 = 𝑧)
15	8, 14	sylan2b 285	. . . 4 ⊢ ((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ [𝑧 / 𝑥]∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)) → 𝑥 = 𝑧)
16	15	gen2 1427	. . 3 ⊢ ∀𝑥∀𝑧((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ [𝑧 / 𝑥]∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)) → 𝑥 = 𝑧)
17	16	jctr 313	. 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 1509	. . 3 ⊢ Ⅎ𝑧∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)
19	18	eu2 2044	. 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 133	1 ⊢ (∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) → ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1330  Ⅎwnf 1437  ∃wex 1469  [wsb 1736  ∃!weu 2000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003

This theorem is referenced by:  zfnuleu  4060