Theorem mosubopt 5489

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 2191	. . 3 ⊢ Ⅎ𝑦∀𝑦∀𝑧∃*𝑥𝜑
2		nfe1 2190	. . . 4 ⊢ Ⅎ𝑦∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
3	2	nfmov 2588	. . 3 ⊢ Ⅎ𝑦∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
4		nfa1 2191	. . . . 5 ⊢ Ⅎ𝑧∀𝑧∃*𝑥𝜑
5		nfe1 2190	. . . . . . 7 ⊢ Ⅎ𝑧∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
6	5	nfex 2358	. . . . . 6 ⊢ Ⅎ𝑧∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
7	6	nfmov 2588	. . . . 5 ⊢ Ⅎ𝑧∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
8		copsexgw 5467	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
9	8	mobidv 2577	. . . . . . 7 ⊢ (𝐴 = ⟨𝑦, 𝑧⟩ → (∃*𝑥𝜑 ↔ ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
10	9	biimpcd 252	. . . . . 6 ⊢ (∃*𝑥𝜑 → (𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
11	10	sps 2225	. . . . 5 ⊢ (∀𝑧∃*𝑥𝜑 → (𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
12	4, 7, 11	exlimd 2258	. . . 4 ⊢ (∀𝑧∃*𝑥𝜑 → (∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
13	12	sps 2225	. . 3 ⊢ (∀𝑦∀𝑧∃*𝑥𝜑 → (∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
14	1, 3, 13	exlimd 2258	. 2 ⊢ (∀𝑦∀𝑧∃*𝑥𝜑 → (∃𝑦∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
15		simpl 488	. . . . 5 ⊢ ((𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → 𝐴 = ⟨𝑦, 𝑧⟩)
16	15	2eximi 1869	. . . 4 ⊢ (∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑦∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩)
17	16	exlimiv 1965	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑦∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩)
18		nexmo 2569	. . 3 ⊢ (¬ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
19	17, 18	nsyl5 160	. 2 ⊢ (¬ ∃𝑦∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
20	14, 19	pm2.61d1 182	1 ⊢ (∀𝑦∀𝑧∃*𝑥𝜑 → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812  ∃*wmo 2565  ⟨cop 4840

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841

This theorem is used by:  mosubop  5490  funoprabg  7572