Theorem mosubopt 4330

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 1417	. . 3 ⊢ Ⅎy∀y∀z∃*xφ
2		nfe1 1367	. . . 4 ⊢ Ⅎy∃y∃z(A = ⟨y, z⟩ ∧ φ)
3	2	nfmo 1903	. . 3 ⊢ Ⅎy∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)
4		nfa1 1417	. . . . 5 ⊢ Ⅎz∀z∃*xφ
5		nfe1 1367	. . . . . . 7 ⊢ Ⅎz∃z(A = ⟨y, z⟩ ∧ φ)
6	5	nfex 1512	. . . . . 6 ⊢ Ⅎz∃y∃z(A = ⟨y, z⟩ ∧ φ)
7	6	nfmo 1903	. . . . 5 ⊢ Ⅎz∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)
8		copsexg 3954	. . . . . . . 8 ⊢ (A = ⟨y, z⟩ → (φ ↔ ∃y∃z(A = ⟨y, z⟩ ∧ φ)))
9	8	mobidv 1919	. . . . . . 7 ⊢ (A = ⟨y, z⟩ → (∃*xφ ↔ ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)))
10	9	biimpcd 148	. . . . . 6 ⊢ (∃*xφ → (A = ⟨y, z⟩ → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)))
11	10	sps 1413	. . . . 5 ⊢ (∀z∃*xφ → (A = ⟨y, z⟩ → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)))
12	4, 7, 11	exlimd 1472	. . . 4 ⊢ (∀z∃*xφ → (∃z A = ⟨y, z⟩ → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)))
13	12	sps 1413	. . 3 ⊢ (∀y∀z∃*xφ → (∃z A = ⟨y, z⟩ → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)))
14	1, 3, 13	exlimd 1472	. 2 ⊢ (∀y∀z∃*xφ → (∃y∃z A = ⟨y, z⟩ → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)))
15		moanimv 1958	. . 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		ax-ia1 99	. . . . . 6 ⊢ ((A = ⟨y, z⟩ ∧ φ) → A = ⟨y, z⟩)
17	16	2eximi 1476	. . . . 5 ⊢ (∃y∃z(A = ⟨y, z⟩ ∧ φ) → ∃y∃z A = ⟨y, z⟩)
18	17	ancri 307	. . . 4 ⊢ (∃y∃z(A = ⟨y, z⟩ ∧ φ) → (∃y∃z A = ⟨y, z⟩ ∧ ∃y∃z(A = ⟨y, z⟩ ∧ φ)))
19	18	moimi 1948	. . 3 ⊢ (∃*x(∃y∃z A = ⟨y, z⟩ ∧ ∃y∃z(A = ⟨y, z⟩ ∧ φ)) → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ))
20	15, 19	sylbir 125	. 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 97  ∀wal 1226   = wceq 1228  ∃wex 1363  ∃*wmo 1884  ⟨cop 3352

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-pow 3900  ax-pr 3917

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-eu 1886  df-mo 1887  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-v 2536  df-un 2898  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355  df-pr 3356  df-op 3358

This theorem is referenced by:  mosubop  4331  funoprabg  5521