Theorem 2eu8 2291

Description: Two equivalent expressions for double existential uniqueness. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!x∃!y using 2eu7 2290. (Contributed by NM, 20-Feb-2005.)

Assertion

Ref	Expression
2eu8	⊢ (∃!x∃!y(∃xφ ∧ ∃yφ) ↔ ∃!x∃!y(∃!xφ ∧ ∃yφ))

Proof of Theorem 2eu8

Step	Hyp	Ref	Expression
1		2eu2 2285	. . 3 ⊢ (∃!x∃yφ → (∃!y∃!xφ ↔ ∃!y∃xφ))
2	1	pm5.32i 618	. 2 ⊢ ((∃!x∃yφ ∧ ∃!y∃!xφ) ↔ (∃!x∃yφ ∧ ∃!y∃xφ))
3		nfeu1 2214	. . . . 5 ⊢ Ⅎx∃!xφ
4	3	nfeu 2220	. . . 4 ⊢ Ⅎx∃!y∃!xφ
5	4	euan 2261	. . 3 ⊢ (∃!x(∃!y∃!xφ ∧ ∃yφ) ↔ (∃!y∃!xφ ∧ ∃!x∃yφ))
6		ancom 437	. . . . . 6 ⊢ ((∃!xφ ∧ ∃yφ) ↔ (∃yφ ∧ ∃!xφ))
7	6	eubii 2213	. . . . 5 ⊢ (∃!y(∃!xφ ∧ ∃yφ) ↔ ∃!y(∃yφ ∧ ∃!xφ))
8		nfe1 1732	. . . . . 6 ⊢ Ⅎy∃yφ
9	8	euan 2261	. . . . 5 ⊢ (∃!y(∃yφ ∧ ∃!xφ) ↔ (∃yφ ∧ ∃!y∃!xφ))
10		ancom 437	. . . . 5 ⊢ ((∃yφ ∧ ∃!y∃!xφ) ↔ (∃!y∃!xφ ∧ ∃yφ))
11	7, 9, 10	3bitri 262	. . . 4 ⊢ (∃!y(∃!xφ ∧ ∃yφ) ↔ (∃!y∃!xφ ∧ ∃yφ))
12	11	eubii 2213	. . 3 ⊢ (∃!x∃!y(∃!xφ ∧ ∃yφ) ↔ ∃!x(∃!y∃!xφ ∧ ∃yφ))
13		ancom 437	. . 3 ⊢ ((∃!x∃yφ ∧ ∃!y∃!xφ) ↔ (∃!y∃!xφ ∧ ∃!x∃yφ))
14	5, 12, 13	3bitr4ri 269	. 2 ⊢ ((∃!x∃yφ ∧ ∃!y∃!xφ) ↔ ∃!x∃!y(∃!xφ ∧ ∃yφ))
15		2eu7 2290	. 2 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ ∃!x∃!y(∃xφ ∧ ∃yφ))
16	2, 14, 15	3bitr3ri 267	1 ⊢ (∃!x∃!y(∃xφ ∧ ∃yφ) ↔ ∃!x∃!y(∃!xφ ∧ ∃yφ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  ∃!weu 2204

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)