Theorem 2moex 2641

Description: Double quantification with "at most one". Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker 2moexv 2628 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 2147	. . 3 ⊢ Ⅎ𝑦∃𝑦𝜑
2	1	nfmo 2561	. 2 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑
3		19.8a 2174	. . 3 ⊢ (𝜑 → ∃𝑦𝜑)
4	3	moimi 2544	. 2 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥𝜑)
5	2, 4	alrimi 2206	1 ⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1539  ∃wex 1781  ∃*wmo 2537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2371

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-mo 2539

This theorem is referenced by:  2eu2  2653