Theorem reuuni2 1956

Description: ∪{x ∈ A∣φ} is the explicit representation of 'the unique element in A such that φ.'

Hypothesis

Ref	Expression
reuuni2.1	⊢ (x = B → (φ ↔ ψ))

Assertion

Ref	Expression
reuuni2	⊢ ((B ∈ A ∧ ∃!x ∈ A φ) → (ψ ↔ ∪{x ∈ A∣φ} = B))

Distinct variable group(s):   x,A   x,B   ψ,x

Proof of Theorem reuuni2

Step	Hyp	Ref	Expression
1		ax-17 925	. . 3 ⊢ (y ∈ B → ∀x y ∈ B)
2		ax-17 925	. . . . 5 ⊢ (B ∈ A → ∀x B ∈ A)
3		hbreu1 1307	. . . . 5 ⊢ (∃!x ∈ A φ → ∀x∃!x ∈ A φ)
4	2, 3	hban 704	. . . 4 ⊢ ((B ∈ A ∧ ∃!x ∈ A φ) → ∀x(B ∈ A ∧ ∃!x ∈ A φ))
5		ax-17 925	. . . . 5 ⊢ (ψ → ∀xψ)
6		hbrab1 1310	. . . . . . 7 ⊢ (y ∈ {x ∈ A∣φ} → ∀x y ∈ {x ∈ A∣φ})
7	6	hbuni 1925	. . . . . 6 ⊢ (y ∈ ∪{x ∈ A∣φ} → ∀x y ∈ ∪{x ∈ A∣φ})
8	7, 1	hbeq 1171	. . . . 5 ⊢ (∪{x ∈ A∣φ} = B → ∀x∪{x ∈ A∣φ} = B)
9	5, 8	hbbi 705	. . . 4 ⊢ ((ψ ↔ ∪{x ∈ A∣φ} = B) → ∀x(ψ ↔ ∪{x ∈ A∣φ} = B))
10	4, 9	hbim 702	. . 3 ⊢ (((B ∈ A ∧ ∃!x ∈ A φ) → (ψ ↔ ∪{x ∈ A∣φ} = B)) → ∀x((B ∈ A ∧ ∃!x ∈ A φ) → (ψ ↔ ∪{x ∈ A∣φ} = B)))
11		eleq1 1149	. . . . 5 ⊢ (x = B → (x ∈ A ↔ B ∈ A))
12	11	anbi1d 469	. . . 4 ⊢ (x = B → ((x ∈ A ∧ ∃!x ∈ A φ) ↔ (B ∈ A ∧ ∃!x ∈ A φ)))
13		reuuni2.1	. . . . 5 ⊢ (x = B → (φ ↔ ψ))
14		cleq2 1110	. . . . 5 ⊢ (x = B → (∪{x ∈ A∣φ} = x ↔ ∪{x ∈ A∣φ} = B))
15	13, 14	bibi12d 477	. . . 4 ⊢ (x = B → ((φ ↔ ∪{x ∈ A∣φ} = x) ↔ (ψ ↔ ∪{x ∈ A∣φ} = B)))
16	12, 15	imbi12d 474	. . 3 ⊢ (x = B → (((x ∈ A ∧ ∃!x ∈ A φ) → (φ ↔ ∪{x ∈ A∣φ} = x)) ↔ ((B ∈ A ∧ ∃!x ∈ A φ) → (ψ ↔ ∪{x ∈ A∣φ} = B))))
17		reuuni1 1955	. . 3 ⊢ ((x ∈ A ∧ ∃!x ∈ A φ) → (φ ↔ ∪{x ∈ A∣φ} = x))
18	1, 10, 16, 17	vtoclgf 1382	. 2 ⊢ (B ∈ A → ((B ∈ A ∧ ∃!x ∈ A φ) → (ψ ↔ ∪{x ∈ A∣φ} = B)))
19	18	anabsi5 377	1 ⊢ ((B ∈ A ∧ ∃!x ∈ A φ) → (ψ ↔ ∪{x ∈ A∣φ} = B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃!wreu 1203  {crab 1204  ∪cuni 1919

This theorem is referenced by:  reuuni3 1958  supub 2160  suplub 2161  supsn 2168  pjpj0 5259

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-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-reu 1207  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-uni 1920

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