Theorem reu6 3026

Description: A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.)

Assertion

Ref	Expression
reu6	⊢ (∃!x ∈ A φ ↔ ∃y ∈ A ∀x ∈ A (φ ↔ x = y))

Distinct variable groups:   x,y,A   φ,y

Allowed substitution hint:   φ(x)

Proof of Theorem reu6

Step	Hyp	Ref	Expression
1		df-reu 2622	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
2		19.28v 1895	. . . . 5 ⊢ (∀x(y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) ↔ (y ∈ A ∧ ∀x(x ∈ A → (φ ↔ x = y))))
3		eleq1 2413	. . . . . . . . . . . 12 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
4		sbequ12 1919	. . . . . . . . . . . 12 ⊢ (x = y → (φ ↔ [y / x]φ))
5	3, 4	anbi12d 691	. . . . . . . . . . 11 ⊢ (x = y → ((x ∈ A ∧ φ) ↔ (y ∈ A ∧ [y / x]φ)))
6		eqeq1 2359	. . . . . . . . . . 11 ⊢ (x = y → (x = y ↔ y = y))
7	5, 6	bibi12d 312	. . . . . . . . . 10 ⊢ (x = y → (((x ∈ A ∧ φ) ↔ x = y) ↔ ((y ∈ A ∧ [y / x]φ) ↔ y = y)))
8		eqid 2353	. . . . . . . . . . . 12 ⊢ y = y
9	8	tbt 333	. . . . . . . . . . 11 ⊢ ((y ∈ A ∧ [y / x]φ) ↔ ((y ∈ A ∧ [y / x]φ) ↔ y = y))
10		simpl 443	. . . . . . . . . . 11 ⊢ ((y ∈ A ∧ [y / x]φ) → y ∈ A)
11	9, 10	sylbir 204	. . . . . . . . . 10 ⊢ (((y ∈ A ∧ [y / x]φ) ↔ y = y) → y ∈ A)
12	7, 11	syl6bi 219	. . . . . . . . 9 ⊢ (x = y → (((x ∈ A ∧ φ) ↔ x = y) → y ∈ A))
13	12	spimv 1990	. . . . . . . 8 ⊢ (∀x((x ∈ A ∧ φ) ↔ x = y) → y ∈ A)
14		bi1 178	. . . . . . . . . . . 12 ⊢ (((x ∈ A ∧ φ) ↔ x = y) → ((x ∈ A ∧ φ) → x = y))
15	14	expdimp 426	. . . . . . . . . . 11 ⊢ ((((x ∈ A ∧ φ) ↔ x = y) ∧ x ∈ A) → (φ → x = y))
16		bi2 189	. . . . . . . . . . . . 13 ⊢ (((x ∈ A ∧ φ) ↔ x = y) → (x = y → (x ∈ A ∧ φ)))
17		simpr 447	. . . . . . . . . . . . 13 ⊢ ((x ∈ A ∧ φ) → φ)
18	16, 17	syl6 29	. . . . . . . . . . . 12 ⊢ (((x ∈ A ∧ φ) ↔ x = y) → (x = y → φ))
19	18	adantr 451	. . . . . . . . . . 11 ⊢ ((((x ∈ A ∧ φ) ↔ x = y) ∧ x ∈ A) → (x = y → φ))
20	15, 19	impbid 183	. . . . . . . . . 10 ⊢ ((((x ∈ A ∧ φ) ↔ x = y) ∧ x ∈ A) → (φ ↔ x = y))
21	20	ex 423	. . . . . . . . 9 ⊢ (((x ∈ A ∧ φ) ↔ x = y) → (x ∈ A → (φ ↔ x = y)))
22	21	sps 1754	. . . . . . . 8 ⊢ (∀x((x ∈ A ∧ φ) ↔ x = y) → (x ∈ A → (φ ↔ x = y)))
23	13, 22	jca 518	. . . . . . 7 ⊢ (∀x((x ∈ A ∧ φ) ↔ x = y) → (y ∈ A ∧ (x ∈ A → (φ ↔ x = y))))
24	23	a5i 1789	. . . . . 6 ⊢ (∀x((x ∈ A ∧ φ) ↔ x = y) → ∀x(y ∈ A ∧ (x ∈ A → (φ ↔ x = y))))
25		bi1 178	. . . . . . . . . . 11 ⊢ ((φ ↔ x = y) → (φ → x = y))
26	25	imim2i 13	. . . . . . . . . 10 ⊢ ((x ∈ A → (φ ↔ x = y)) → (x ∈ A → (φ → x = y)))
27	26	imp3a 420	. . . . . . . . 9 ⊢ ((x ∈ A → (φ ↔ x = y)) → ((x ∈ A ∧ φ) → x = y))
28	27	adantl 452	. . . . . . . 8 ⊢ ((y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) → ((x ∈ A ∧ φ) → x = y))
29		eleq1a 2422	. . . . . . . . . . . 12 ⊢ (y ∈ A → (x = y → x ∈ A))
30	29	adantr 451	. . . . . . . . . . 11 ⊢ ((y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) → (x = y → x ∈ A))
31	30	imp 418	. . . . . . . . . 10 ⊢ (((y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) ∧ x = y) → x ∈ A)
32		bi2 189	. . . . . . . . . . . . . 14 ⊢ ((φ ↔ x = y) → (x = y → φ))
33	32	imim2i 13	. . . . . . . . . . . . 13 ⊢ ((x ∈ A → (φ ↔ x = y)) → (x ∈ A → (x = y → φ)))
34	33	com23 72	. . . . . . . . . . . 12 ⊢ ((x ∈ A → (φ ↔ x = y)) → (x = y → (x ∈ A → φ)))
35	34	imp 418	. . . . . . . . . . 11 ⊢ (((x ∈ A → (φ ↔ x = y)) ∧ x = y) → (x ∈ A → φ))
36	35	adantll 694	. . . . . . . . . 10 ⊢ (((y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) ∧ x = y) → (x ∈ A → φ))
37	31, 36	jcai 522	. . . . . . . . 9 ⊢ (((y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) ∧ x = y) → (x ∈ A ∧ φ))
38	37	ex 423	. . . . . . . 8 ⊢ ((y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) → (x = y → (x ∈ A ∧ φ)))
39	28, 38	impbid 183	. . . . . . 7 ⊢ ((y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) → ((x ∈ A ∧ φ) ↔ x = y))
40	39	alimi 1559	. . . . . 6 ⊢ (∀x(y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) → ∀x((x ∈ A ∧ φ) ↔ x = y))
41	24, 40	impbii 180	. . . . 5 ⊢ (∀x((x ∈ A ∧ φ) ↔ x = y) ↔ ∀x(y ∈ A ∧ (x ∈ A → (φ ↔ x = y))))
42		df-ral 2620	. . . . . 6 ⊢ (∀x ∈ A (φ ↔ x = y) ↔ ∀x(x ∈ A → (φ ↔ x = y)))
43	42	anbi2i 675	. . . . 5 ⊢ ((y ∈ A ∧ ∀x ∈ A (φ ↔ x = y)) ↔ (y ∈ A ∧ ∀x(x ∈ A → (φ ↔ x = y))))
44	2, 41, 43	3bitr4i 268	. . . 4 ⊢ (∀x((x ∈ A ∧ φ) ↔ x = y) ↔ (y ∈ A ∧ ∀x ∈ A (φ ↔ x = y)))
45	44	exbii 1582	. . 3 ⊢ (∃y∀x((x ∈ A ∧ φ) ↔ x = y) ↔ ∃y(y ∈ A ∧ ∀x ∈ A (φ ↔ x = y)))
46		df-eu 2208	. . 3 ⊢ (∃!x(x ∈ A ∧ φ) ↔ ∃y∀x((x ∈ A ∧ φ) ↔ x = y))
47		df-rex 2621	. . 3 ⊢ (∃y ∈ A ∀x ∈ A (φ ↔ x = y) ↔ ∃y(y ∈ A ∧ ∀x ∈ A (φ ↔ x = y)))
48	45, 46, 47	3bitr4i 268	. 2 ⊢ (∃!x(x ∈ A ∧ φ) ↔ ∃y ∈ A ∀x ∈ A (φ ↔ x = y))
49	1, 48	bitri 240	1 ⊢ (∃!x ∈ A φ ↔ ∃y ∈ A ∀x ∈ A (φ ↔ x = y))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  [wsb 1648   ∈ wcel 1710  ∃!weu 2204  ∀wral 2615  ∃wrex 2616  ∃!wreu 2617

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-cleq 2346  df-clel 2349  df-ral 2620  df-rex 2621  df-reu 2622

This theorem is referenced by:  reu3  3027  reu6i  3028  reu8  3033