Theorem mosubopt 5507

Description: "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)

Assertion

Ref	Expression
mosubopt	⊢ (∀𝑦∀𝑧∃*𝑥𝜑 → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))

Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem mosubopt

Step	Hyp	Ref	Expression
1		nfa1 2141	. . 3 ⊢ Ⅎ𝑦∀𝑦∀𝑧∃*𝑥𝜑
2		nfe1 2140	. . . 4 ⊢ Ⅎ𝑦∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
3	2	nfmov 2550	. . 3 ⊢ Ⅎ𝑦∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
4		nfa1 2141	. . . . 5 ⊢ Ⅎ𝑧∀𝑧∃*𝑥𝜑
5		nfe1 2140	. . . . . . 7 ⊢ Ⅎ𝑧∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
6	5	nfex 2313	. . . . . 6 ⊢ Ⅎ𝑧∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
7	6	nfmov 2550	. . . . 5 ⊢ Ⅎ𝑧∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
8		copsexgw 5487	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
9	8	mobidv 2539	. . . . . . 7 ⊢ (𝐴 = ⟨𝑦, 𝑧⟩ → (∃*𝑥𝜑 ↔ ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
10	9	biimpcd 248	. . . . . 6 ⊢ (∃*𝑥𝜑 → (𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
11	10	sps 2174	. . . . 5 ⊢ (∀𝑧∃*𝑥𝜑 → (𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
12	4, 7, 11	exlimd 2207	. . . 4 ⊢ (∀𝑧∃*𝑥𝜑 → (∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
13	12	sps 2174	. . 3 ⊢ (∀𝑦∀𝑧∃*𝑥𝜑 → (∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
14	1, 3, 13	exlimd 2207	. 2 ⊢ (∀𝑦∀𝑧∃*𝑥𝜑 → (∃𝑦∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
15		simpl 482	. . . . 5 ⊢ ((𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → 𝐴 = ⟨𝑦, 𝑧⟩)
16	15	2eximi 1831	. . . 4 ⊢ (∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑦∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩)
17	16	exlimiv 1926	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑦∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩)
18		nexmo 2531	. . 3 ⊢ (¬ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
19	17, 18	nsyl5 159	. 2 ⊢ (¬ ∃𝑦∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
20	14, 19	pm2.61d1 180	1 ⊢ (∀𝑦∀𝑧∃*𝑥𝜑 → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1532   = wceq 1534  ∃wex 1774  ∃*wmo 2528  ⟨cop 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632

This theorem is referenced by:  mosubop  5508  funoprabg  7536

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