Theorem 2moex 2642

Description: Double quantification with "at most one". Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker 2moexv 2629 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 2562	. 2 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑
3		19.8a 2174	. . 3 ⊢ (𝜑 → ∃𝑦𝜑)
4	3	moimi 2545	. 2 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥𝜑)
5	2, 4	alrimi 2206	1 ⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1782  ∃*wmo 2538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-mo 2540

This theorem is referenced by:  2eu2  2654