Theorem reu6 2724

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 2307	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
2		19.28v 1777	. . . . 5 ⊢ (∀x(y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) ↔ (y ∈ A ∧ ∀x(x ∈ A → (φ ↔ x = y))))
3		eleq1 2097	. . . . . . . . . . . 12 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
4		sbequ12 1651	. . . . . . . . . . . 12 ⊢ (x = y → (φ ↔ [y / x]φ))
5	3, 4	anbi12d 442	. . . . . . . . . . 11 ⊢ (x = y → ((x ∈ A ∧ φ) ↔ (y ∈ A ∧ [y / x]φ)))
6		equequ1 1595	. . . . . . . . . . 11 ⊢ (x = y → (x = y ↔ y = y))
7	5, 6	bibi12d 224	. . . . . . . . . 10 ⊢ (x = y → (((x ∈ A ∧ φ) ↔ x = y) ↔ ((y ∈ A ∧ [y / x]φ) ↔ y = y)))
8		equid 1586	. . . . . . . . . . . 12 ⊢ y = y
9	8	tbt 236	. . . . . . . . . . 11 ⊢ ((y ∈ A ∧ [y / x]φ) ↔ ((y ∈ A ∧ [y / x]φ) ↔ y = y))
10		simpl 102	. . . . . . . . . . 11 ⊢ ((y ∈ A ∧ [y / x]φ) → y ∈ A)
11	9, 10	sylbir 125	. . . . . . . . . 10 ⊢ (((y ∈ A ∧ [y / x]φ) ↔ y = y) → y ∈ A)
12	7, 11	syl6bi 152	. . . . . . . . 9 ⊢ (x = y → (((x ∈ A ∧ φ) ↔ x = y) → y ∈ A))
13	12	spimv 1689	. . . . . . . 8 ⊢ (∀x((x ∈ A ∧ φ) ↔ x = y) → y ∈ A)
14		bi1 111	. . . . . . . . . . . 12 ⊢ (((x ∈ A ∧ φ) ↔ x = y) → ((x ∈ A ∧ φ) → x = y))
15	14	expdimp 246	. . . . . . . . . . 11 ⊢ ((((x ∈ A ∧ φ) ↔ x = y) ∧ x ∈ A) → (φ → x = y))
16		bi2 121	. . . . . . . . . . . . 13 ⊢ (((x ∈ A ∧ φ) ↔ x = y) → (x = y → (x ∈ A ∧ φ)))
17		simpr 103	. . . . . . . . . . . . 13 ⊢ ((x ∈ A ∧ φ) → φ)
18	16, 17	syl6 29	. . . . . . . . . . . 12 ⊢ (((x ∈ A ∧ φ) ↔ x = y) → (x = y → φ))
19	18	adantr 261	. . . . . . . . . . 11 ⊢ ((((x ∈ A ∧ φ) ↔ x = y) ∧ x ∈ A) → (x = y → φ))
20	15, 19	impbid 120	. . . . . . . . . 10 ⊢ ((((x ∈ A ∧ φ) ↔ x = y) ∧ x ∈ A) → (φ ↔ x = y))
21	20	ex 108	. . . . . . . . 9 ⊢ (((x ∈ A ∧ φ) ↔ x = y) → (x ∈ A → (φ ↔ x = y)))
22	21	sps 1427	. . . . . . . 8 ⊢ (∀x((x ∈ A ∧ φ) ↔ x = y) → (x ∈ A → (φ ↔ x = y)))
23	13, 22	jca 290	. . . . . . 7 ⊢ (∀x((x ∈ A ∧ φ) ↔ x = y) → (y ∈ A ∧ (x ∈ A → (φ ↔ x = y))))
24	23	a5i 1432	. . . . . 6 ⊢ (∀x((x ∈ A ∧ φ) ↔ x = y) → ∀x(y ∈ A ∧ (x ∈ A → (φ ↔ x = y))))
25		bi1 111	. . . . . . . . . . 11 ⊢ ((φ ↔ x = y) → (φ → x = y))
26	25	imim2i 12	. . . . . . . . . 10 ⊢ ((x ∈ A → (φ ↔ x = y)) → (x ∈ A → (φ → x = y)))
27	26	impd 242	. . . . . . . . 9 ⊢ ((x ∈ A → (φ ↔ x = y)) → ((x ∈ A ∧ φ) → x = y))
28	27	adantl 262	. . . . . . . 8 ⊢ ((y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) → ((x ∈ A ∧ φ) → x = y))
29		eleq1a 2106	. . . . . . . . . . . 12 ⊢ (y ∈ A → (x = y → x ∈ A))
30	29	adantr 261	. . . . . . . . . . 11 ⊢ ((y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) → (x = y → x ∈ A))
31	30	imp 115	. . . . . . . . . 10 ⊢ (((y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) ∧ x = y) → x ∈ A)
32		bi2 121	. . . . . . . . . . . . . 14 ⊢ ((φ ↔ x = y) → (x = y → φ))
33	32	imim2i 12	. . . . . . . . . . . . 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 115	. . . . . . . . . . 11 ⊢ (((x ∈ A → (φ ↔ x = y)) ∧ x = y) → (x ∈ A → φ))
36	35	adantll 445	. . . . . . . . . 10 ⊢ (((y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) ∧ x = y) → (x ∈ A → φ))
37	31, 36	jcai 294	. . . . . . . . 9 ⊢ (((y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) ∧ x = y) → (x ∈ A ∧ φ))
38	37	ex 108	. . . . . . . 8 ⊢ ((y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) → (x = y → (x ∈ A ∧ φ)))
39	28, 38	impbid 120	. . . . . . 7 ⊢ ((y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) → ((x ∈ A ∧ φ) ↔ x = y))
40	39	alimi 1341	. . . . . 6 ⊢ (∀x(y ∈ A ∧ (x ∈ A → (φ ↔ x = y))) → ∀x((x ∈ A ∧ φ) ↔ x = y))
41	24, 40	impbii 117	. . . . 5 ⊢ (∀x((x ∈ A ∧ φ) ↔ x = y) ↔ ∀x(y ∈ A ∧ (x ∈ A → (φ ↔ x = y))))
42		df-ral 2305	. . . . . 6 ⊢ (∀x ∈ A (φ ↔ x = y) ↔ ∀x(x ∈ A → (φ ↔ x = y)))
43	42	anbi2i 430	. . . . 5 ⊢ ((y ∈ A ∧ ∀x ∈ A (φ ↔ x = y)) ↔ (y ∈ A ∧ ∀x(x ∈ A → (φ ↔ x = y))))
44	2, 41, 43	3bitr4i 201	. . . 4 ⊢ (∀x((x ∈ A ∧ φ) ↔ x = y) ↔ (y ∈ A ∧ ∀x ∈ A (φ ↔ x = y)))
45	44	exbii 1493	. . 3 ⊢ (∃y∀x((x ∈ A ∧ φ) ↔ x = y) ↔ ∃y(y ∈ A ∧ ∀x ∈ A (φ ↔ x = y)))
46		df-eu 1900	. . 3 ⊢ (∃!x(x ∈ A ∧ φ) ↔ ∃y∀x((x ∈ A ∧ φ) ↔ x = y))
47		df-rex 2306	. . 3 ⊢ (∃y ∈ A ∀x ∈ A (φ ↔ x = y) ↔ ∃y(y ∈ A ∧ ∀x ∈ A (φ ↔ x = y)))
48	45, 46, 47	3bitr4i 201	. 2 ⊢ (∃!x(x ∈ A ∧ φ) ↔ ∃y ∈ A ∀x ∈ A (φ ↔ x = y))
49	1, 48	bitri 173	1 ⊢ (∃!x ∈ A φ ↔ ∃y ∈ A ∀x ∈ A (φ ↔ x = y))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  ∃wex 1378   ∈ wcel 1390  [wsb 1642  ∃!weu 1897  ∀wral 2300  ∃wrex 2301  ∃!wreu 2302

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-cleq 2030  df-clel 2033  df-ral 2305  df-rex 2306  df-reu 2307

This theorem is referenced by:  reu3  2725  reu6i  2726  reu8  2731