Theorem 2moex 2543

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 2027	. . 3 ⊢ Ⅎ𝑦∃𝑦𝜑
2	1	nfmo 2487	. 2 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑
3		19.8a 2052	. . 3 ⊢ (𝜑 → ∃𝑦𝜑)
4	3	moimi 2520	. 2 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥𝜑)
5	2, 4	alrimi 2082	1 ⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1481  ∃wex 1704  ∃*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  2eu2  2554  2eu5  2557