Theorem 2moex 2638

Description: Double quantification with "at most one". Usage of this theorem is discouraged because it depends on ax-13 2375. Use the weaker 2moexv 2625 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 2148	. . 3 ⊢ Ⅎ𝑦∃𝑦𝜑
2	1	nfmo 2560	. 2 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑
3		19.8a 2179	. . 3 ⊢ (𝜑 → ∃𝑦𝜑)
4	3	moimi 2543	. 2 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥𝜑)
5	2, 4	alrimi 2211	1 ⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1776  ∃*wmo 2536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-mo 2538

This theorem is referenced by:  2eu2  2651