Theorem moabex 1868

Description: "At most one" existence implies a class abstraction exists.

Assertion

Ref	Expression
moabex	⊢ (∃*xφ → {x∣φ} ∈ V)

Proof of Theorem moabex

Step	Hyp	Ref	Expression
1		ax-17 925	. . 3 ⊢ (φ → ∀yφ)
2	1	mo2 1026	. 2 ⊢ (∃*xφ ↔ ∃y∀x(φ → x = y))
3		df-sn 1811	. . . . . 6 ⊢ {y} = {x∣x = y}
4		snex 1859	. . . . . 6 ⊢ {y} ∈ V
5	3, 4	eqeltrr 1160	. . . . 5 ⊢ {x∣x = y} ∈ V
6		pm3.26 256	. . . . . 6 ⊢ ((x = y ∧ φ) → x = y)
7	6	ss2abi 1552	. . . . 5 ⊢ {x∣(x = y ∧ φ)} ⊆ {x∣x = y}
8	5, 7	ssexi 1701	. . . 4 ⊢ {x∣(x = y ∧ φ)} ∈ V
9		hba1 698	. . . . . 6 ⊢ (∀x(φ → x = y) → ∀x∀x(φ → x = y))
10		pm4.71 481	. . . . . . . . 9 ⊢ ((φ → x = y) ↔ (φ ↔ (φ ∧ x = y)))
11	10	biimp 133	. . . . . . . 8 ⊢ ((φ → x = y) → (φ ↔ (φ ∧ x = y)))
12	11	a4s 682	. . . . . . 7 ⊢ (∀x(φ → x = y) → (φ ↔ (φ ∧ x = y)))
13		ancom 333	. . . . . . 7 ⊢ ((φ ∧ x = y) ↔ (x = y ∧ φ))
14	12, 13	syl6bb 414	. . . . . 6 ⊢ (∀x(φ → x = y) → (φ ↔ (x = y ∧ φ)))
15	9, 14	biabd 1182	. . . . 5 ⊢ (∀x(φ → x = y) → {x∣φ} = {x∣(x = y ∧ φ)})
16	15	eleq1d 1155	. . . 4 ⊢ (∀x(φ → x = y) → ({x∣φ} ∈ V ↔ {x∣(x = y ∧ φ)} ∈ V))
17	8, 16	mpbiri 169	. . 3 ⊢ (∀x(φ → x = y) → {x∣φ} ∈ V)
18	17	19.23aiv 952	. 2 ⊢ (∃y∀x(φ → x = y) → {x∣φ} ∈ V)
19	2, 18	sylbi 174	1 ⊢ (∃*xφ → {x∣φ} ∈ V)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  ∃*wmo 1008  {cab 1090   ∈ wcel 1092  Vcvv 1348  {csn 1808

This theorem is referenced by:  euabex 1869  supex 2157  fvex 2838

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811

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