Theorem 2moex 2665

Description: Double quantification with "at most one". Usage of this theorem is discouraged because it depends on ax-13 2401. Use the weaker 2moexv 2652 when possible. (Contributed by NM, 3-Dec-2001.) (New usage is discouraged.)

Assertion

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

Proof of Theorem 2moex

Step	Hyp	Ref	Expression
1		nfe1 2186	. . 3 ⊢ Ⅎ𝑦∃𝑦𝜑
2	1	nfmo 2587	. 2 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑
3		19.8a 2217	. . 3 ⊢ (𝜑 → ∃𝑦𝜑)
4	3	moimi 2570	. 2 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥𝜑)
5	2, 4	alrimi 2249	1 ⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1811  ∃*wmo 2562

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-11 2193  ax-12 2212  ax-13 2401

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-mo 2564

This theorem is used by:  2eu2  2677