Theorem mosubopt 4739

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 1563	. . 3 ⊢ Ⅎ𝑦∀𝑦∀𝑧∃*𝑥𝜑
2		nfe1 1518	. . . 4 ⊢ Ⅎ𝑦∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
3	2	nfmo 2073	. . 3 ⊢ Ⅎ𝑦∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
4		nfa1 1563	. . . . 5 ⊢ Ⅎ𝑧∀𝑧∃*𝑥𝜑
5		nfe1 1518	. . . . . . 7 ⊢ Ⅎ𝑧∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
6	5	nfex 1659	. . . . . 6 ⊢ Ⅎ𝑧∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
7	6	nfmo 2073	. . . . 5 ⊢ Ⅎ𝑧∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
8		copsexg 4287	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
9	8	mobidv 2089	. . . . . . 7 ⊢ (𝐴 = ⟨𝑦, 𝑧⟩ → (∃*𝑥𝜑 ↔ ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
10	9	biimpcd 159	. . . . . 6 ⊢ (∃*𝑥𝜑 → (𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
11	10	sps 1559	. . . . 5 ⊢ (∀𝑧∃*𝑥𝜑 → (𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
12	4, 7, 11	exlimd 1619	. . . 4 ⊢ (∀𝑧∃*𝑥𝜑 → (∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
13	12	sps 1559	. . 3 ⊢ (∀𝑦∀𝑧∃*𝑥𝜑 → (∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
14	1, 3, 13	exlimd 1619	. 2 ⊢ (∀𝑦∀𝑧∃*𝑥𝜑 → (∃𝑦∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
15		moanimv 2128	. . 3 ⊢ (∃*𝑥(∃𝑦∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ ∧ ∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)) ↔ (∃𝑦∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
16		simpl 109	. . . . . 6 ⊢ ((𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → 𝐴 = ⟨𝑦, 𝑧⟩)
17	16	2eximi 1623	. . . . 5 ⊢ (∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑦∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩)
18	17	ancri 324	. . . 4 ⊢ (∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → (∃𝑦∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ ∧ ∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
19	18	moimi 2118	. . 3 ⊢ (∃*𝑥(∃𝑦∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ ∧ ∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)) → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
20	15, 19	sylbir 135	. 2 ⊢ ((∃𝑦∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)) → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
21	14, 20	syl 14	1 ⊢ (∀𝑦∀𝑧∃*𝑥𝜑 → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1370   = wceq 1372  ∃wex 1514  ∃*wmo 2054  ⟨cop 3635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641

This theorem is referenced by:  mosubop  4740  funoprabg  6043