Theorem reu6 3666

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 3286	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		19.28v 1992	. . . . 5 ⊢ (∀𝑥(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
3		eleq1w 2819	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
4		sbequ12 2242	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
5	3, 4	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)))
6		equequ1 2026	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑦 ↔ 𝑦 = 𝑦))
7	5, 6	bibi12d 346	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦)))
8		equid 2013	. . . . . . . . . . . 12 ⊢ 𝑦 = 𝑦
9	8	tbt 370	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝑦 = 𝑦))
10		simpl 484	. . . . . . . . . . 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 253	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → 𝑦 ∈ 𝐴))
13	12	spimvw 1997	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → 𝑦 ∈ 𝐴)
14		ibar 530	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
15	14	bibi1d 344	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → ((𝜑 ↔ 𝑥 = 𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦)))
16	15	biimprcd 250	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)))
17	16	sps 2176	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)))
18	13, 17	jca 513	. . . . . . 7 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) → (𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
19	18	axc4i 2314	. . . . . 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 412	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
23	22	adantl 483	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
24	3	biimprcd 250	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝑥 ∈ 𝐴))
25	24	adantr 482	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) → (𝑥 = 𝑦 → 𝑥 ∈ 𝐴))
26	25	imp 408	. . . . . . . . . 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 486	. . . . . . . . . . 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 518	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ∧ 𝜑))
32	31	ex 414	. . . . . . . 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 1811	. . . . . 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 3063	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦)))
37	36	anbi2i 624	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ↔ 𝑥 = 𝑦))))
38	2, 35, 37	3bitr4i 303	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)))
39	38	exbii 1848	. . 3 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)))
40		eu6 2572	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦))
41		df-rex 3072	. . 3 ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦)))
42	39, 40, 41	3bitr4i 303	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))
43	1, 42	bitri 275	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397  ∀wal 1537  ∃wex 1779  [wsb 2065   ∈ wcel 2104  ∃!weu 2566  ∀wral 3062  ∃wrex 3071  ∃!wreu 3282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-10 2135  ax-12 2169

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clel 2814  df-ral 3063  df-rex 3072  df-reu 3286

This theorem is referenced by:  reu3  3667  reu6i  3668  reu8  3673  xpf1o  8964  ufileu  23115  isppw2  26309  cusgrfilem2  27868  fgreu  31054  fcnvgreu  31055  gsumhashmul  31361  fourierdlem50  43746