Theorem reu6 3646

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 3110	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		19.28v 1972	. . . . 5 ⊢ (∀𝑥(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
3		eleq1w 2863	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
4		sbequ12 2214	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
5	3, 4	anbi12d 630	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)))
6		equequ1 2007	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑦 ↔ 𝑦 = 𝑦))
7	5, 6	bibi12d 347	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦)))
8		equid 1994	. . . . . . . . . . . 12 ⊢ 𝑦 = 𝑦
9	8	tbt 371	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦))
10		simpl 483	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) → 𝑦 ∈ 𝐴)
11	9, 10	sylbir 236	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦) → 𝑦 ∈ 𝐴)
12	7, 11	syl6bi 254	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → 𝑦 ∈ 𝐴))
13	12	spimvw 1977	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → 𝑦 ∈ 𝐴)
14		ibar 529	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
15	14	bibi1d 345	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → ((𝜑 ↔ 𝑥 = 𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦)))
16	15	biimprcd 251	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)))
17	16	sps 2146	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)))
18	13, 17	jca 512	. . . . . . 7 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → (𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
19	18	axc4i 2302	. . . . . 6 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → ∀𝑥(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
20		biimp 216	. . . . . . . . . . 11 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦))
21	20	imim2i 16	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)) → (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
22	21	impd 411	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
23	22	adantl 482	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
24		eleq1a 2876	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝑥 ∈ 𝐴))
25	24	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → (𝑥 = 𝑦 → 𝑥 ∈ 𝐴))
26	25	imp 407	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥 ∈ 𝐴)
27		biimpr 221	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝑥 = 𝑦 → 𝜑))
28	27	imim2i 16	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)) → (𝑥 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)))
29	28	com23 86	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)) → (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 → 𝜑)))
30	29	imp 407	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)) ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → 𝜑))
31	30	adantll 710	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → 𝜑))
32	26, 31	jcai 517	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ∧ 𝜑))
33	32	ex 413	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ∧ 𝜑)))
34	23, 33	impbid 213	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦))
35	34	alimi 1791	. . . . . 6 ⊢ (∀𝑥(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦))
36	19, 35	impbii 210	. . . . 5 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
37		df-ral 3108	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)))
38	37	anbi2i 622	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
39	2, 36, 38	3bitr4i 304	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)))
40	39	exbii 1827	. . 3 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)))
41		eu6 2615	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦))
42		df-rex 3109	. . 3 ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)))
43	40, 41, 42	3bitr4i 304	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))
44	1, 43	bitri 276	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1518  ∃wex 1759  [wsb 2040   ∈ wcel 2079  ∃!weu 2609  ∀wral 3103  ∃wrex 3104  ∃!wreu 3105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-12 2139  ax-ext 2767

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-cleq 2786  df-clel 2861  df-ral 3108  df-rex 3109  df-reu 3110

This theorem is referenced by:  reu3  3647  reu6i  3648  reu8  3653  xpf1o  8516  ufileu  22199  isppw2  25362  cusgrfilem2  26909  fgreu  30080  fcnvgreu  30081  fourierdlem50  41937