Theorem reu8 3691

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 3388	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
3		reu6 3684	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥))
4		dfbi2 480	. . . . 5 ⊢ ((𝜓 ↔ 𝑦 = 𝑥) ↔ ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)))
5	4	ralbii 3109	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ ∀𝑦 ∈ 𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)))
6		r19.26 3123	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)))
7		ancom 466	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ∧ 𝜑))
8		equcom 2051	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
9	8	imbi2i 339	. . . . . . . . 9 ⊢ ((𝜓 → 𝑥 = 𝑦) ↔ (𝜓 → 𝑦 = 𝑥))
10	9	ralbii 3109	. . . . . . . 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 363	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 → 𝜑)))
13		df-ral 3078	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓)))
14		bi2.04 392	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓)) ↔ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓)))
15	14	albii 1852	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓)))
16		eleq1w 2844	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
17	16, 1	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜓)))
18	17	bicomd 226	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑦 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜑)))
19	18	equcoms 2053	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜑)))
20	19	equsalvw 2037	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓)) ↔ (𝑥 ∈ 𝐴 → 𝜑))
21	13, 15, 20	3bitrri 301	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓))
22	12, 21	bitrdi 290	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)))
23	11, 22	anbi12d 644	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ∧ 𝜑) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓))))
24	7, 23	bitrid 286	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓))))
25	6, 24	bitr4id 293	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))))
26	5, 25	bitrid 286	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))))
27	26	rexbiia 3108	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
28	2, 3, 27	3bitri 300	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  ∃!wreu 3364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-10 2178  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clel 2836  df-ral 3078  df-rex 3088  df-reu 3367

This theorem is used by:  reu8nf  3824  updjud  10015  reusq0  15632  reumodprminv  16982  grpinveu  19185  addsq2reu  27767  2sqreulem1  27773  2sqreunnlem1  27776  grpoideu  31111  grpoinveu  31121  cvmlift3lem2  36085  euoreqb  48178  2reu8i  48182  2reuimp0  48183  paireqne  48592  pgnbgreunbgr  49222  itsclquadeu  49888