Theorem 2eu6 2658

Description: Two equivalent expressions for double existential uniqueness. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Oct-2019.)

Assertion

Ref	Expression
2eu6	⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤   𝜑,𝑧,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2eu6

Step	Hyp	Ref	Expression
1		2eu4 2656	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
2		nfia1 2150	. . . . . 6 ⊢ Ⅎ𝑥(∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
3		nfa1 2148	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))
4		nfv 1917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦 𝑥 = 𝑧
5		simpl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
6	5	imim2i 16	. . . . . . . . . . . . . 14 ⊢ ((𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝜑 → 𝑥 = 𝑧))
7	6	sps 2178	. . . . . . . . . . . . 13 ⊢ (∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝜑 → 𝑥 = 𝑧))
8	3, 4, 7	exlimd 2211	. . . . . . . . . . . 12 ⊢ (∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (∃𝑦𝜑 → 𝑥 = 𝑧))
9		ax12v 2172	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (∃𝑦𝜑 → ∀𝑥(𝑥 = 𝑧 → ∃𝑦𝜑)))
10	8, 9	syli 39	. . . . . . . . . . 11 ⊢ (∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (∃𝑦𝜑 → ∀𝑥(𝑥 = 𝑧 → ∃𝑦𝜑)))
11	10	com12 32	. . . . . . . . . 10 ⊢ (∃𝑦𝜑 → (∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∀𝑥(𝑥 = 𝑧 → ∃𝑦𝜑)))
12	11	spsd 2180	. . . . . . . . 9 ⊢ (∃𝑦𝜑 → (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∀𝑥(𝑥 = 𝑧 → ∃𝑦𝜑)))
13		nfs1v 2153	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜑
14		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
15	14	imim2i 16	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝜑 → 𝑦 = 𝑤))
16		sbequ1 2240	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → (𝜑 → [𝑤 / 𝑦]𝜑))
17	15, 16	syli 39	. . . . . . . . . . . . . 14 ⊢ ((𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝜑 → [𝑤 / 𝑦]𝜑))
18	17	sps 2178	. . . . . . . . . . . . 13 ⊢ (∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝜑 → [𝑤 / 𝑦]𝜑))
19	3, 13, 18	exlimd 2211	. . . . . . . . . . . 12 ⊢ (∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (∃𝑦𝜑 → [𝑤 / 𝑦]𝜑))
20	19	imim2d 57	. . . . . . . . . . 11 ⊢ (∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ((𝑥 = 𝑧 → ∃𝑦𝜑) → (𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑)))
21	20	al2imi 1818	. . . . . . . . . 10 ⊢ (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (∀𝑥(𝑥 = 𝑧 → ∃𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑)))
22		sb6 2088	. . . . . . . . . . 11 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑))
23		2sb6 2089	. . . . . . . . . . 11 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
24	22, 23	bitr3i 276	. . . . . . . . . 10 ⊢ (∀𝑥(𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑) ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
25	21, 24	syl6ib 250	. . . . . . . . 9 ⊢ (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (∀𝑥(𝑥 = 𝑧 → ∃𝑦𝜑) → ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑)))
26	12, 25	sylcom 30	. . . . . . . 8 ⊢ (∃𝑦𝜑 → (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑)))
27	26	ancld 551	. . . . . . 7 ⊢ (∃𝑦𝜑 → (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ∧ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))))
28		2albiim 1893	. . . . . . 7 ⊢ (∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ∧ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑)))
29	27, 28	syl6ibr 251	. . . . . 6 ⊢ (∃𝑦𝜑 → (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
30	2, 29	exlimi 2210	. . . . 5 ⊢ (∃𝑥∃𝑦𝜑 → (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
31	30	2eximdv 1922	. . . 4 ⊢ (∃𝑥∃𝑦𝜑 → (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
32	31	imp 407	. . 3 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
33		biimpr 219	. . . . . . 7 ⊢ ((𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
34	33	2alimi 1815	. . . . . 6 ⊢ (∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
35	34	2eximi 1838	. . . . 5 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
36		2exsb 2358	. . . . 5 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
37	35, 36	sylibr 233	. . . 4 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∃𝑥∃𝑦𝜑)
38		biimp 214	. . . . . 6 ⊢ ((𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
39	38	2alimi 1815	. . . . 5 ⊢ (∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
40	39	2eximi 1838	. . . 4 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
41	37, 40	jca 512	. . 3 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
42	32, 41	impbii 208	. 2 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
43	1, 42	bitri 274	1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537  ∃wex 1782  [wsb 2067  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569

This theorem is referenced by: (None)