Theorem reu6 3362

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 2903	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		19.28v 1896	. . . . 5 ⊢ (∀𝑥(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
3		eleq1 2676	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
4		sbequ12 2097	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
5	3, 4	anbi12d 743	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)))
6		equequ1 1939	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑦 ↔ 𝑦 = 𝑦))
7	5, 6	bibi12d 334	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦)))
8		equid 1926	. . . . . . . . . . . 12 ⊢ 𝑦 = 𝑦
9	8	tbt 358	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦))
10		simpl 472	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) → 𝑦 ∈ 𝐴)
11	9, 10	sylbir 224	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦) → 𝑦 ∈ 𝐴)
12	7, 11	syl6bi 242	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → 𝑦 ∈ 𝐴))
13	12	spimv 2245	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → 𝑦 ∈ 𝐴)
14		ibar 524	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
15	14	bibi1d 332	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → ((𝜑 ↔ 𝑥 = 𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦)))
16	15	biimprcd 239	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)))
17	16	sps 2043	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)))
18	13, 17	jca 553	. . . . . . 7 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → (𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
19	18	axc4i 2116	. . . . . 6 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → ∀𝑥(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
20		biimp 204	. . . . . . . . . . 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 446	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
23	22	adantl 481	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
24		eleq1a 2683	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝑥 ∈ 𝐴))
25	24	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → (𝑥 = 𝑦 → 𝑥 ∈ 𝐴))
26	25	imp 444	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥 ∈ 𝐴)
27		biimpr 209	. . . . . . . . . . . . . 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 84	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)) → (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 → 𝜑)))
30	29	imp 444	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)) ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → 𝜑))
31	30	adantll 746	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → 𝜑))
32	26, 31	jcai 557	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ∧ 𝜑))
33	32	ex 449	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ∧ 𝜑)))
34	23, 33	impbid 201	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦))
35	34	alimi 1730	. . . . . 6 ⊢ (∀𝑥(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦))
36	19, 35	impbii 198	. . . . 5 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
37		df-ral 2901	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)))
38	37	anbi2i 726	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
39	2, 36, 38	3bitr4i 291	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)))
40	39	exbii 1764	. . 3 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)))
41		df-eu 2462	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦))
42		df-rex 2902	. . 3 ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)))
43	40, 41, 42	3bitr4i 291	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))
44	1, 43	bitri 263	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  [wsb 1867   ∈ wcel 1977  ∃!weu 2458  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-cleq 2603  df-clel 2606  df-ral 2901  df-rex 2902  df-reu 2903

This theorem is referenced by:  reu3  3363  reu6i  3364  reu8  3369  xpf1o  8007  ufileu  21533  isppw2  24641  cusgrafilem2  26008  fgreu  28854  fcnvgreu  28855  fourierdlem50  39049  cusgrfilem2  40672