Theorem 2euex 2276

Description: Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
2euex	⊢ (∃!x∃yφ → ∃y∃!xφ)

Proof of Theorem 2euex

Step	Hyp	Ref	Expression
1		eu5 2242	. 2 ⊢ (∃!x∃yφ ↔ (∃x∃yφ ∧ ∃*x∃yφ))
2		excom 1741	. . . 4 ⊢ (∃x∃yφ ↔ ∃y∃xφ)
3		nfe1 1732	. . . . . 6 ⊢ Ⅎy∃yφ
4	3	nfmo 2221	. . . . 5 ⊢ Ⅎy∃*x∃yφ
5		19.8a 1756	. . . . . . 7 ⊢ (φ → ∃yφ)
6	5	moimi 2251	. . . . . 6 ⊢ (∃*x∃yφ → ∃*xφ)
7		df-mo 2209	. . . . . 6 ⊢ (∃*xφ ↔ (∃xφ → ∃!xφ))
8	6, 7	sylib 188	. . . . 5 ⊢ (∃*x∃yφ → (∃xφ → ∃!xφ))
9	4, 8	eximd 1770	. . . 4 ⊢ (∃*x∃yφ → (∃y∃xφ → ∃y∃!xφ))
10	2, 9	syl5bi 208	. . 3 ⊢ (∃*x∃yφ → (∃x∃yφ → ∃y∃!xφ))
11	10	impcom 419	. 2 ⊢ ((∃x∃yφ ∧ ∃*x∃yφ) → ∃y∃!xφ)
12	1, 11	sylbi 187	1 ⊢ (∃!x∃yφ → ∃y∃!xφ)

Syntax hints:   → wi 4   ∧ wa 358  ∃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:  2exeu  2281