Theorem reu8 3724

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 3452	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
3		reu6 3717	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥))
4		dfbi2 477	. . . . 5 ⊢ ((𝜓 ↔ 𝑦 = 𝑥) ↔ ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)))
5	4	ralbii 3165	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ ∀𝑦 ∈ 𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)))
6		ancom 463	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ∧ 𝜑))
7		equcom 2021	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
8	7	imbi2i 338	. . . . . . . . 9 ⊢ ((𝜓 → 𝑥 = 𝑦) ↔ (𝜓 → 𝑦 = 𝑥))
9	8	ralbii 3165	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥))
10	9	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥)))
11		biimt 363	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 → 𝜑)))
12		df-ral 3143	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓)))
13		bi2.04 391	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓)) ↔ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓)))
14	13	albii 1816	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓)))
15		eleq1w 2895	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
16	15, 1	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜓)))
17	16	bicomd 225	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑦 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜑)))
18	17	equcoms 2023	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜑)))
19	18	equsalvw 2006	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓)) ↔ (𝑥 ∈ 𝐴 → 𝜑))
20	12, 14, 19	3bitrri 300	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓))
21	11, 20	syl6bb 289	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)))
22	10, 21	anbi12d 632	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ∧ 𝜑) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓))))
23	6, 22	syl5bb 285	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓))))
24		r19.26 3170	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)))
25	23, 24	syl6rbbr 292	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))))
26	5, 25	syl5bb 285	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))))
27	26	rexbiia 3246	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
28	2, 3, 27	3bitri 299	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  ∃!wreu 3140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-10 2141  ax-11 2156  ax-12 2172

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clel 2893  df-ral 3143  df-rex 3144  df-reu 3145

This theorem is referenced by:  reu8nf  3860  updjud  9357  reusq0  14816  reumodprminv  16135  grpinveu  18132  addsq2reu  26010  2sqreulem1  26016  2sqreunnlem1  26019  grpoideu  28280  grpoinveu  28290  cvmlift3lem2  32562  euoreqb  43301  2reu8i  43305  2reuimp0  43306  paireqne  43666  itsclquadeu  44757