Theorem 2eu3 2286

Description: Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)

Assertion

Ref	Expression
2eu3	⊢ (∀x∀y(∃*xφ ∨ ∃*yφ) → ((∃!x∃!yφ ∧ ∃!y∃!xφ) ↔ (∃!x∃yφ ∧ ∃!y∃xφ)))

Proof of Theorem 2eu3

Step	Hyp	Ref	Expression
1		nfmo1 2215	. . . . 5 ⊢ Ⅎy∃*yφ
2	1	19.31 1876	. . . 4 ⊢ (∀y(∃*xφ ∨ ∃*yφ) ↔ (∀y∃*xφ ∨ ∃*yφ))
3	2	albii 1566	. . 3 ⊢ (∀x∀y(∃*xφ ∨ ∃*yφ) ↔ ∀x(∀y∃*xφ ∨ ∃*yφ))
4		nfmo1 2215	. . . . 5 ⊢ Ⅎx∃*xφ
5	4	nfal 1842	. . . 4 ⊢ Ⅎx∀y∃*xφ
6	5	19.32 1875	. . 3 ⊢ (∀x(∀y∃*xφ ∨ ∃*yφ) ↔ (∀y∃*xφ ∨ ∀x∃*yφ))
7	3, 6	bitri 240	. 2 ⊢ (∀x∀y(∃*xφ ∨ ∃*yφ) ↔ (∀y∃*xφ ∨ ∀x∃*yφ))
8		2eu1 2284	. . . . . . 7 ⊢ (∀y∃*xφ → (∃!y∃!xφ ↔ (∃!y∃xφ ∧ ∃!x∃yφ)))
9	8	biimpd 198	. . . . . 6 ⊢ (∀y∃*xφ → (∃!y∃!xφ → (∃!y∃xφ ∧ ∃!x∃yφ)))
10		ancom 437	. . . . . 6 ⊢ ((∃!y∃xφ ∧ ∃!x∃yφ) ↔ (∃!x∃yφ ∧ ∃!y∃xφ))
11	9, 10	syl6ib 217	. . . . 5 ⊢ (∀y∃*xφ → (∃!y∃!xφ → (∃!x∃yφ ∧ ∃!y∃xφ)))
12	11	adantld 453	. . . 4 ⊢ (∀y∃*xφ → ((∃!x∃!yφ ∧ ∃!y∃!xφ) → (∃!x∃yφ ∧ ∃!y∃xφ)))
13		2eu1 2284	. . . . . 6 ⊢ (∀x∃*yφ → (∃!x∃!yφ ↔ (∃!x∃yφ ∧ ∃!y∃xφ)))
14	13	biimpd 198	. . . . 5 ⊢ (∀x∃*yφ → (∃!x∃!yφ → (∃!x∃yφ ∧ ∃!y∃xφ)))
15	14	adantrd 454	. . . 4 ⊢ (∀x∃*yφ → ((∃!x∃!yφ ∧ ∃!y∃!xφ) → (∃!x∃yφ ∧ ∃!y∃xφ)))
16	12, 15	jaoi 368	. . 3 ⊢ ((∀y∃*xφ ∨ ∀x∃*yφ) → ((∃!x∃!yφ ∧ ∃!y∃!xφ) → (∃!x∃yφ ∧ ∃!y∃xφ)))
17		2exeu 2281	. . . 4 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) → ∃!x∃!yφ)
18		2exeu 2281	. . . . 5 ⊢ ((∃!y∃xφ ∧ ∃!x∃yφ) → ∃!y∃!xφ)
19	18	ancoms 439	. . . 4 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) → ∃!y∃!xφ)
20	17, 19	jca 518	. . 3 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) → (∃!x∃!yφ ∧ ∃!y∃!xφ))
21	16, 20	impbid1 194	. 2 ⊢ ((∀y∃*xφ ∨ ∀x∃*yφ) → ((∃!x∃!yφ ∧ ∃!y∃!xφ) ↔ (∃!x∃yφ ∧ ∃!y∃xφ)))
22	7, 21	sylbi 187	1 ⊢ (∀x∀y(∃*xφ ∨ ∃*yφ) → ((∃!x∃!yφ ∧ ∃!y∃!xφ) ↔ (∃!x∃yφ ∧ ∃!y∃xφ)))

Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541  ∃!weu 2204  ∃*wmo 2205

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

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-mo 2209

This theorem is referenced by: (None)