Theorem moexex 2632

Description: "At most one" double quantification. Usage of this theorem is discouraged because it depends on ax-13 2369. Use the version moexexvw 2622 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.) Factor out common proof lines with moexexvw 2622. (Revised by Wolf Lammen, 2-Oct-2023.) (New usage is discouraged.)

Hypothesis

Ref	Expression
moexex.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
moexex	⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))

Proof of Theorem moexex

Step	Hyp	Ref	Expression
1		moexex.1	. 2 ⊢ Ⅎ𝑦𝜑
2	1	nfmo 2554	. 2 ⊢ Ⅎ𝑦∃*𝑥𝜑
3		nfe1 2145	. . 3 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓)
4	3	nfmo 2554	. 2 ⊢ Ⅎ𝑥∃*𝑦∃𝑥(𝜑 ∧ 𝜓)
5	1, 2, 4	moexexlem 2620	1 ⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 394  ∀wal 1537  ∃wex 1779  Ⅎwnf 1783  ∃*wmo 2530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-11 2152  ax-12 2169  ax-13 2369

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-mo 2532

This theorem is referenced by:  moexexv  2633  2moswap  2638