Theorem 2moswap 2279

Description: A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.)

Assertion

Ref	Expression
2moswap	⊢ (∀x∃*yφ → (∃*x∃yφ → ∃*y∃xφ))

Proof of Theorem 2moswap

Step	Hyp	Ref	Expression
1		nfe1 1732	. . . 4 ⊢ Ⅎy∃yφ
2	1	moexex 2273	. . 3 ⊢ ((∃*x∃yφ ∧ ∀x∃*yφ) → ∃*y∃x(∃yφ ∧ φ))
3	2	expcom 424	. 2 ⊢ (∀x∃*yφ → (∃*x∃yφ → ∃*y∃x(∃yφ ∧ φ)))
4		19.8a 1756	. . . . 5 ⊢ (φ → ∃yφ)
5	4	pm4.71ri 614	. . . 4 ⊢ (φ ↔ (∃yφ ∧ φ))
6	5	exbii 1582	. . 3 ⊢ (∃xφ ↔ ∃x(∃yφ ∧ φ))
7	6	mobii 2240	. 2 ⊢ (∃*y∃xφ ↔ ∃*y∃x(∃yφ ∧ φ))
8	3, 7	syl6ibr 218	1 ⊢ (∀x∃*yφ → (∃*x∃yφ → ∃*y∃xφ))

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  ∃*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:  2euswap  2280  2eu1  2284