Theorem 2eu7 2290

Description: Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)

Assertion

Ref	Expression
2eu7	⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ ∃!x∃!y(∃xφ ∧ ∃yφ))

Proof of Theorem 2eu7

Step	Hyp	Ref	Expression
1		nfe1 1732	. . . 4 ⊢ Ⅎx∃xφ
2	1	nfeu 2220	. . 3 ⊢ Ⅎx∃!y∃xφ
3	2	euan 2261	. 2 ⊢ (∃!x(∃!y∃xφ ∧ ∃yφ) ↔ (∃!y∃xφ ∧ ∃!x∃yφ))
4		ancom 437	. . . . 5 ⊢ ((∃xφ ∧ ∃yφ) ↔ (∃yφ ∧ ∃xφ))
5	4	eubii 2213	. . . 4 ⊢ (∃!y(∃xφ ∧ ∃yφ) ↔ ∃!y(∃yφ ∧ ∃xφ))
6		nfe1 1732	. . . . 5 ⊢ Ⅎy∃yφ
7	6	euan 2261	. . . 4 ⊢ (∃!y(∃yφ ∧ ∃xφ) ↔ (∃yφ ∧ ∃!y∃xφ))
8		ancom 437	. . . 4 ⊢ ((∃yφ ∧ ∃!y∃xφ) ↔ (∃!y∃xφ ∧ ∃yφ))
9	5, 7, 8	3bitri 262	. . 3 ⊢ (∃!y(∃xφ ∧ ∃yφ) ↔ (∃!y∃xφ ∧ ∃yφ))
10	9	eubii 2213	. 2 ⊢ (∃!x∃!y(∃xφ ∧ ∃yφ) ↔ ∃!x(∃!y∃xφ ∧ ∃yφ))
11		ancom 437	. 2 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ (∃!y∃xφ ∧ ∃!x∃yφ))
12	3, 10, 11	3bitr4ri 269	1 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ ∃!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:  2eu8  2291