Theorem reu6 3687

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 3352	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		19.28v 1994	. . . . 5 ⊢ (∀𝑥(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
3		eleq1w 2815	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
4		sbequ12 2243	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
5	3, 4	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)))
6		equequ1 2028	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑦 ↔ 𝑦 = 𝑦))
7	5, 6	bibi12d 345	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦)))
8		equid 2015	. . . . . . . . . . . 12 ⊢ 𝑦 = 𝑦
9	8	tbt 369	. . . . . . . . . . 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 234	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦) → 𝑦 ∈ 𝐴)
12	7, 11	syl6bi 252	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → 𝑦 ∈ 𝐴))
13	12	spimvw 1999	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → 𝑦 ∈ 𝐴)
14		ibar 529	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
15	14	bibi1d 343	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → ((𝜑 ↔ 𝑥 = 𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦)))
16	15	biimprcd 249	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)))
17	16	sps 2178	. . . . . . . 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 2315	. . . . . 6 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → ∀𝑥(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
20		biimp 214	. . . . . . . . . . 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	3	biimprcd 249	. . . . . . . . . . . 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		simplr 767	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)))
28		simpr 485	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
29		biimpr 219	. . . . . . . . . . 11 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝑥 = 𝑦 → 𝜑))
30	27, 28, 29	syl6ci 71	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → 𝜑))
31	26, 30	jcai 517	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ∧ 𝜑))
32	31	ex 413	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ∧ 𝜑)))
33	23, 32	impbid 211	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦))
34	33	alimi 1813	. . . . . 6 ⊢ (∀𝑥(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦))
35	19, 34	impbii 208	. . . . 5 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
36		df-ral 3061	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)))
37	36	anbi2i 623	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
38	2, 35, 37	3bitr4i 302	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)))
39	38	exbii 1850	. . 3 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)))
40		eu6 2567	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦))
41		df-rex 3070	. . 3 ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)))
42	39, 40, 41	3bitr4i 302	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))
43	1, 42	bitri 274	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539  ∃wex 1781  [wsb 2067   ∈ wcel 2106  ∃!weu 2561  ∀wral 3060  ∃wrex 3069  ∃!wreu 3349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-10 2137  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clel 2809  df-ral 3061  df-rex 3070  df-reu 3352

This theorem is referenced by:  reu3  3688  reu6i  3689  reu8  3694  xpf1o  9090  ufileu  23307  isppw2  26501  cusgrfilem2  28467  fgreu  31655  fcnvgreu  31656  gsumhashmul  31968  fourierdlem50  44517