Theorem moexex 2705

Description: "At most one" double quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.)

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		nfmo1 2643	. . . 4 ⊢ Ⅎ𝑥∃*𝑥𝜑
2		nfa1 2195	. . . . 5 ⊢ Ⅎ𝑥∀𝑥∃*𝑦𝜓
3		nfe1 2194	. . . . . 6 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓)
4	3	nfmo 2649	. . . . 5 ⊢ Ⅎ𝑥∃*𝑦∃𝑥(𝜑 ∧ 𝜓)
5	2, 4	nfim 1987	. . . 4 ⊢ Ⅎ𝑥(∀𝑥∃*𝑦𝜓 → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))
6		moexex.1	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
7	6	nfmo 2649	. . . . . 6 ⊢ Ⅎ𝑦∃*𝑥𝜑
8		mopick 2699	. . . . . . . 8 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
9	8	ex 399	. . . . . . 7 ⊢ (∃*𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) → (𝜑 → 𝜓)))
10	9	com23 86	. . . . . 6 ⊢ (∃*𝑥𝜑 → (𝜑 → (∃𝑥(𝜑 ∧ 𝜓) → 𝜓)))
11	7, 6, 10	alrimd 2251	. . . . 5 ⊢ (∃*𝑥𝜑 → (𝜑 → ∀𝑦(∃𝑥(𝜑 ∧ 𝜓) → 𝜓)))
12		moim 2682	. . . . . 6 ⊢ (∀𝑦(∃𝑥(𝜑 ∧ 𝜓) → 𝜓) → (∃*𝑦𝜓 → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓)))
13	12	spsd 2222	. . . . 5 ⊢ (∀𝑦(∃𝑥(𝜑 ∧ 𝜓) → 𝜓) → (∀𝑥∃*𝑦𝜓 → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓)))
14	11, 13	syl6 35	. . . 4 ⊢ (∃*𝑥𝜑 → (𝜑 → (∀𝑥∃*𝑦𝜓 → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))))
15	1, 5, 14	exlimd 2254	. . 3 ⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 → (∀𝑥∃*𝑦𝜓 → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))))
16	6	nfex 2330	. . . . . . 7 ⊢ Ⅎ𝑦∃𝑥𝜑
17		exsimpl 1956	. . . . . . 7 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)
18	16, 17	exlimi 2253	. . . . . 6 ⊢ (∃𝑦∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)
19		exmo 2657	. . . . . . 7 ⊢ (∃𝑦∃𝑥(𝜑 ∧ 𝜓) ∨ ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))
20	19	ori 879	. . . . . 6 ⊢ (¬ ∃𝑦∃𝑥(𝜑 ∧ 𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))
21	18, 20	nsyl4 157	. . . . 5 ⊢ (¬ ∃*𝑦∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)
22	21	con1i 146	. . . 4 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))
23	22	a1d 25	. . 3 ⊢ (¬ ∃𝑥𝜑 → (∀𝑥∃*𝑦𝜓 → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓)))
24	15, 23	pm2.61d1 172	. 2 ⊢ (∃*𝑥𝜑 → (∀𝑥∃*𝑦𝜓 → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓)))
25	24	imp 395	1 ⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384  ∀wal 1635  ∃wex 1859  Ⅎwnf 1863  ∃*wmo 2631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-eu 2634  df-mo 2635

This theorem is referenced by:  moexexv  2706  2moswap  2711