Theorem reu8 3755

Description: Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)

Hypothesis

Ref	Expression
rmo4.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
reu8	⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem reu8

Step	Hyp	Ref	Expression
1		rmo4.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	cbvreuvw 3412	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
3		reu6 3748	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥))
4		dfbi2 474	. . . . 5 ⊢ ((𝜓 ↔ 𝑦 = 𝑥) ↔ ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)))
5	4	ralbii 3099	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ ∀𝑦 ∈ 𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)))
6		r19.26 3117	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)))
7		ancom 460	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ∧ 𝜑))
8		equcom 2017	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
9	8	imbi2i 336	. . . . . . . . 9 ⊢ ((𝜓 → 𝑥 = 𝑦) ↔ (𝜓 → 𝑦 = 𝑥))
10	9	ralbii 3099	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥))
11	10	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥)))
12		biimt 360	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 → 𝜑)))
13		df-ral 3068	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓)))
14		bi2.04 387	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓)) ↔ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓)))
15	14	albii 1817	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓)))
16		eleq1w 2827	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
17	16, 1	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜓)))
18	17	bicomd 223	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑦 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜑)))
19	18	equcoms 2019	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜑)))
20	19	equsalvw 2003	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓)) ↔ (𝑥 ∈ 𝐴 → 𝜑))
21	13, 15, 20	3bitrri 298	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓))
22	12, 21	bitrdi 287	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)))
23	11, 22	anbi12d 631	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ∧ 𝜑) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓))))
24	7, 23	bitrid 283	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓))))
25	6, 24	bitr4id 290	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))))
26	5, 25	bitrid 283	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))))
27	26	rexbiia 3098	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
28	2, 3, 27	3bitri 297	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  ∃!wreu 3386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-10 2141  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clel 2819  df-ral 3068  df-rex 3077  df-reu 3389

This theorem is referenced by:  reu8nf  3899  updjud  10003  reusq0  15511  reumodprminv  16851  grpinveu  19014  addsq2reu  27502  2sqreulem1  27508  2sqreunnlem1  27511  grpoideu  30541  grpoinveu  30551  cvmlift3lem2  35288  euoreqb  47024  2reu8i  47028  2reuimp0  47029  paireqne  47385  itsclquadeu  48511