Theorem reu6 3683

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 3366	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		19.28v 2028	. . . . 5 ⊢ (∀𝑥(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
3		eleq1w 2843	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
4		sbequ12 2286	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
5	3, 4	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)))
6		equequ1 2057	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑦 ↔ 𝑦 = 𝑦))
7	5, 6	bibi12d 348	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦)))
8		equid 2044	. . . . . . . . . . . 12 ⊢ 𝑦 = 𝑦
9	8	tbt 372	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦))
10		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) → 𝑦 ∈ 𝐴)
11	9, 10	sylbir 238	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦) → 𝑦 ∈ 𝐴)
12	7, 11	biimtrdi 256	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → 𝑦 ∈ 𝐴))
13	12	spimvw 2018	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → 𝑦 ∈ 𝐴)
14		ibar 538	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
15	14	bibi1d 346	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → ((𝜑 ↔ 𝑥 = 𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦)))
16	15	biimprcd 253	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)))
17	16	sps 2221	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)))
18	13, 17	jca 521	. . . . . . 7 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → (𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
19	18	axc4i 2352	. . . . . 6 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → ∀𝑥(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
20		biimp 218	. . . . . . . . . . 11 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦))
21	20	imim2i 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)) → (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
22	21	impd 416	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
23	22	adantl 487	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
24	3	biimprcd 253	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝑥 ∈ 𝐴))
25	24	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → (𝑥 = 𝑦 → 𝑥 ∈ 𝐴))
26	25	imp 412	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥 ∈ 𝐴)
27		simplr 781	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)))
28		simpr 490	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
29		biimpr 223	. . . . . . . . . . 11 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝑥 = 𝑦 → 𝜑))
30	27, 28, 29	syl6ci 72	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → 𝜑))
31	26, 30	jcai 526	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ∧ 𝜑))
32	31	ex 418	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ∧ 𝜑)))
33	23, 32	impbid 215	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦))
34	33	alimi 1843	. . . . . 6 ⊢ (∀𝑥(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦))
35	19, 34	impbii 212	. . . . 5 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
36		df-ral 3077	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)))
37	36	anbi2i 635	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
38	2, 35, 37	3bitr4i 306	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)))
39	38	exbii 1880	. . 3 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)))
40		eu6 2599	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦))
41		df-rex 3087	. . 3 ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)))
42	39, 40, 41	3bitr4i 306	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))
43	1, 42	bitri 278	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567  ∃wex 1811  [wsb 2098   ∈ wcel 2144  ∃!weu 2593  ∀wral 3076  ∃wrex 3086  ∃!wreu 3363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clel 2835  df-ral 3077  df-rex 3087  df-reu 3366

This theorem is used by:  reu3  3684  reu6i  3685  reu8  3690  xpf1o  9158  ufileu  24238  isppw2  27442  cusgrfilem2  30037  fgreu  33265  fcnvgreu  33266  gsumhashmul  33628  aks6d1c7  43235  modelac8prim  45982  fourierdlem50  47166