Theorem reu8 3033

Description: Restricted unique existence 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	cbvreuv 2838	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!y ∈ A ψ)
3		reu6 3026	. 2 ⊢ (∃!y ∈ A ψ ↔ ∃x ∈ A ∀y ∈ A (ψ ↔ y = x))
4		dfbi2 609	. . . . 5 ⊢ ((ψ ↔ y = x) ↔ ((ψ → y = x) ∧ (y = x → ψ)))
5	4	ralbii 2639	. . . 4 ⊢ (∀y ∈ A (ψ ↔ y = x) ↔ ∀y ∈ A ((ψ → y = x) ∧ (y = x → ψ)))
6		ancom 437	. . . . . 6 ⊢ ((φ ∧ ∀y ∈ A (ψ → x = y)) ↔ (∀y ∈ A (ψ → x = y) ∧ φ))
7		equcom 1680	. . . . . . . . . 10 ⊢ (x = y ↔ y = x)
8	7	imbi2i 303	. . . . . . . . 9 ⊢ ((ψ → x = y) ↔ (ψ → y = x))
9	8	ralbii 2639	. . . . . . . 8 ⊢ (∀y ∈ A (ψ → x = y) ↔ ∀y ∈ A (ψ → y = x))
10	9	a1i 10	. . . . . . 7 ⊢ (x ∈ A → (∀y ∈ A (ψ → x = y) ↔ ∀y ∈ A (ψ → y = x)))
11		biimt 325	. . . . . . . 8 ⊢ (x ∈ A → (φ ↔ (x ∈ A → φ)))
12		df-ral 2620	. . . . . . . . 9 ⊢ (∀y ∈ A (y = x → ψ) ↔ ∀y(y ∈ A → (y = x → ψ)))
13		bi2.04 350	. . . . . . . . . 10 ⊢ ((y ∈ A → (y = x → ψ)) ↔ (y = x → (y ∈ A → ψ)))
14	13	albii 1566	. . . . . . . . 9 ⊢ (∀y(y ∈ A → (y = x → ψ)) ↔ ∀y(y = x → (y ∈ A → ψ)))
15		vex 2863	. . . . . . . . . 10 ⊢ x ∈ V
16		eleq1 2413	. . . . . . . . . . . . 13 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
17	16, 1	imbi12d 311	. . . . . . . . . . . 12 ⊢ (x = y → ((x ∈ A → φ) ↔ (y ∈ A → ψ)))
18	17	bicomd 192	. . . . . . . . . . 11 ⊢ (x = y → ((y ∈ A → ψ) ↔ (x ∈ A → φ)))
19	18	equcoms 1681	. . . . . . . . . 10 ⊢ (y = x → ((y ∈ A → ψ) ↔ (x ∈ A → φ)))
20	15, 19	ceqsalv 2886	. . . . . . . . 9 ⊢ (∀y(y = x → (y ∈ A → ψ)) ↔ (x ∈ A → φ))
21	12, 14, 20	3bitrri 263	. . . . . . . 8 ⊢ ((x ∈ A → φ) ↔ ∀y ∈ A (y = x → ψ))
22	11, 21	syl6bb 252	. . . . . . 7 ⊢ (x ∈ A → (φ ↔ ∀y ∈ A (y = x → ψ)))
23	10, 22	anbi12d 691	. . . . . 6 ⊢ (x ∈ A → ((∀y ∈ A (ψ → x = y) ∧ φ) ↔ (∀y ∈ A (ψ → y = x) ∧ ∀y ∈ A (y = x → ψ))))
24	6, 23	syl5bb 248	. . . . 5 ⊢ (x ∈ A → ((φ ∧ ∀y ∈ A (ψ → x = y)) ↔ (∀y ∈ A (ψ → y = x) ∧ ∀y ∈ A (y = x → ψ))))
25		r19.26 2747	. . . . 5 ⊢ (∀y ∈ A ((ψ → y = x) ∧ (y = x → ψ)) ↔ (∀y ∈ A (ψ → y = x) ∧ ∀y ∈ A (y = x → ψ)))
26	24, 25	syl6rbbr 255	. . . 4 ⊢ (x ∈ A → (∀y ∈ A ((ψ → y = x) ∧ (y = x → ψ)) ↔ (φ ∧ ∀y ∈ A (ψ → x = y))))
27	5, 26	syl5bb 248	. . 3 ⊢ (x ∈ A → (∀y ∈ A (ψ ↔ y = x) ↔ (φ ∧ ∀y ∈ A (ψ → x = y))))
28	27	rexbiia 2648	. 2 ⊢ (∃x ∈ A ∀y ∈ A (ψ ↔ y = x) ↔ ∃x ∈ A (φ ∧ ∀y ∈ A (ψ → x = y)))
29	2, 3, 28	3bitri 262	1 ⊢ (∃!x ∈ A φ ↔ ∃x ∈ A (φ ∧ ∀y ∈ A (ψ → x = y)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616  ∃!wreu 2617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2620  df-rex 2621  df-reu 2622  df-v 2862

This theorem is referenced by: (None)