Theorem 2eu6 2657

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 2655	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
2		nfia1 2156	. . . . . 6 ⊢ Ⅎ𝑥(∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
3		nfa1 2154	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))
4		nfv 1922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦 𝑥 = 𝑧
5		simpl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
6	5	imim2i 16	. . . . . . . . . . . . . 14 ⊢ ((𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝜑 → 𝑥 = 𝑧))
7	6	sps 2184	. . . . . . . . . . . . 13 ⊢ (∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝜑 → 𝑥 = 𝑧))
8	3, 4, 7	exlimd 2218	. . . . . . . . . . . 12 ⊢ (∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (∃𝑦𝜑 → 𝑥 = 𝑧))
9		ax12v 2178	. . . . . . . . . . . 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 2186	. . . . . . . . 9 ⊢ (∃𝑦𝜑 → (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∀𝑥(𝑥 = 𝑧 → ∃𝑦𝜑)))
13		nfs1v 2159	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜑
14		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
15	14	imim2i 16	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝜑 → 𝑦 = 𝑤))
16		sbequ1 2247	. . . . . . . . . . . . . . 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 2184	. . . . . . . . . . . . 13 ⊢ (∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝜑 → [𝑤 / 𝑦]𝜑))
19	3, 13, 18	exlimd 2218	. . . . . . . . . . . 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 1823	. . . . . . . . . 10 ⊢ (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (∀𝑥(𝑥 = 𝑧 → ∃𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑)))
22		sb6 2093	. . . . . . . . . . 11 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑))
23		2sb6 2094	. . . . . . . . . . 11 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
24	22, 23	bitr3i 280	. . . . . . . . . 10 ⊢ (∀𝑥(𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑) ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
25	21, 24	syl6ib 254	. . . . . . . . 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 554	. . . . . . 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 1898	. . . . . . 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 255	. . . . . 6 ⊢ (∃𝑦𝜑 → (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
30	2, 29	exlimi 2217	. . . . 5 ⊢ (∃𝑥∃𝑦𝜑 → (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
31	30	2eximdv 1927	. . . 4 ⊢ (∃𝑥∃𝑦𝜑 → (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
32	31	imp 410	. . 3 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
33		biimpr 223	. . . . . . 7 ⊢ ((𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
34	33	2alimi 1820	. . . . . 6 ⊢ (∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
35	34	2eximi 1843	. . . . 5 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
36		2exsb 2360	. . . . 5 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
37	35, 36	sylibr 237	. . . 4 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∃𝑥∃𝑦𝜑)
38		biimp 218	. . . . . 6 ⊢ ((𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
39	38	2alimi 1820	. . . . 5 ⊢ (∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
40	39	2eximi 1843	. . . 4 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
41	37, 40	jca 515	. . 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 212	. 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 278	1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541  ∃wex 1787  [wsb 2072  ∃!weu 2567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-10 2143  ax-11 2160  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568

This theorem is referenced by: (None)