Theorem 2eu2 2285

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

Assertion

Ref	Expression
2eu2	⊢ (∃!y∃xφ → (∃!x∃!yφ ↔ ∃!x∃yφ))

Proof of Theorem 2eu2

Step	Hyp	Ref	Expression
1		eumo 2244	. . 3 ⊢ (∃!y∃xφ → ∃*y∃xφ)
2		2moex 2275	. . 3 ⊢ (∃*y∃xφ → ∀x∃*yφ)
3		2eu1 2284	. . . 4 ⊢ (∀x∃*yφ → (∃!x∃!yφ ↔ (∃!x∃yφ ∧ ∃!y∃xφ)))
4		simpl 443	. . . 4 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) → ∃!x∃yφ)
5	3, 4	syl6bi 219	. . 3 ⊢ (∀x∃*yφ → (∃!x∃!yφ → ∃!x∃yφ))
6	1, 2, 5	3syl 18	. 2 ⊢ (∃!y∃xφ → (∃!x∃!yφ → ∃!x∃yφ))
7		2exeu 2281	. . 3 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) → ∃!x∃!yφ)
8	7	expcom 424	. 2 ⊢ (∃!y∃xφ → (∃!x∃yφ → ∃!x∃!yφ))
9	6, 8	impbid 183	1 ⊢ (∃!y∃xφ → (∃!x∃!yφ ↔ ∃!x∃yφ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ 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:  2eu8  2291