Theorem 2moex 2275

Description: Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)

Assertion

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

Proof of Theorem 2moex

Step	Hyp	Ref	Expression
1		nfe1 1732	. . 3 ⊢ Ⅎy∃yφ
2	1	nfmo 2221	. 2 ⊢ Ⅎy∃*x∃yφ
3		19.8a 1756	. . 3 ⊢ (φ → ∃yφ)
4	3	moimi 2251	. 2 ⊢ (∃*x∃yφ → ∃*xφ)
5	2, 4	alrimi 1765	1 ⊢ (∃*x∃yφ → ∀y∃*xφ)

Syntax hints:   → wi 4  ∀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:  2eu2  2285  2eu5  2288