Theorem mosubott 5488

Description: "At most one" remains true inside ordered triple quantification, analogous to mosubopt 5486. (Contributed by BTernaryTau, 8-Sep-2026.)

Assertion

Ref	Expression
mosubott	⊢ (∀𝑥∀𝑦∀𝑧∃*𝑤𝜑 → ∃*𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑))

Distinct variable group:   𝑤,𝐴,𝑥,𝑦,𝑧

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

Proof of Theorem mosubott

Step	Hyp	Ref	Expression
1		nfa1 2191	. . 3 ⊢ Ⅎ𝑥∀𝑥∀𝑦∀𝑧∃*𝑤𝜑
2		nfe1 2190	. . . 4 ⊢ Ⅎ𝑥∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)
3	2	nfmov 2588	. . 3 ⊢ Ⅎ𝑥∃*𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)
4		nfa1 2191	. . . . 5 ⊢ Ⅎ𝑦∀𝑦∀𝑧∃*𝑤𝜑
5		nfe1 2190	. . . . . . 7 ⊢ Ⅎ𝑦∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)
6	5	nfex 2358	. . . . . 6 ⊢ Ⅎ𝑦∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)
7	6	nfmov 2588	. . . . 5 ⊢ Ⅎ𝑦∃*𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)
8		nfa1 2191	. . . . . . 7 ⊢ Ⅎ𝑧∀𝑧∃*𝑤𝜑
9		nfe1 2190	. . . . . . . . . 10 ⊢ Ⅎ𝑧∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)
10	9	nfex 2358	. . . . . . . . 9 ⊢ Ⅎ𝑧∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)
11	10	nfex 2358	. . . . . . . 8 ⊢ Ⅎ𝑧∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)
12	11	nfmov 2588	. . . . . . 7 ⊢ Ⅎ𝑧∃*𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)
13		cotsexgw 5467	. . . . . . . . . 10 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)))
14	13	mobidv 2577	. . . . . . . . 9 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (∃*𝑤𝜑 ↔ ∃*𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)))
15	14	biimpcd 252	. . . . . . . 8 ⊢ (∃*𝑤𝜑 → (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃*𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)))
16	15	sps 2225	. . . . . . 7 ⊢ (∀𝑧∃*𝑤𝜑 → (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃*𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)))
17	8, 12, 16	exlimd 2258	. . . . . 6 ⊢ (∀𝑧∃*𝑤𝜑 → (∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃*𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)))
18	17	sps 2225	. . . . 5 ⊢ (∀𝑦∀𝑧∃*𝑤𝜑 → (∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃*𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)))
19	4, 7, 18	exlimd 2258	. . . 4 ⊢ (∀𝑦∀𝑧∃*𝑤𝜑 → (∃𝑦∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃*𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)))
20	19	sps 2225	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧∃*𝑤𝜑 → (∃𝑦∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃*𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)))
21	1, 3, 20	exlimd 2258	. 2 ⊢ (∀𝑥∀𝑦∀𝑧∃*𝑤𝜑 → (∃𝑥∃𝑦∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃*𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)))
22		exsimpl 1901	. . . . 5 ⊢ (∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
23	22	2eximi 1869	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
24	23	exlimiv 1965	. . 3 ⊢ (∃𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
25		nexmo 2569	. . 3 ⊢ (¬ ∃𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑) → ∃*𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑))
26	24, 25	nsyl5 160	. 2 ⊢ (¬ ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃*𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑))
27	21, 26	pm2.61d1 182	1 ⊢ (∀𝑥∀𝑦∀𝑧∃*𝑤𝜑 → ∃*𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812  ∃*wmo 2565  ⟨cotp 4861

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 5252  ax-pr 5395

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-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-ot 4862

This theorem is used by:  funmpt3  7711  mpt3fvd  7712