Theorem mosubop 1911

Description: "At most one" remains true inside order pair quantification.

Hypothesis

Ref	Expression
mosubop.1	⊢ ∃*xφ

Assertion

Ref	Expression
mosubop	⊢ ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)

Distinct variable group(s):   x,y,z,A

Proof of Theorem mosubop

Step	Hyp	Ref	Expression
1		hbe1 709	. . . 4 ⊢ (∃y∃z(A = ⟨y, z⟩ ∧ φ) → ∀y∃y∃z(A = ⟨y, z⟩ ∧ φ))
2	1	hbmo 1033	. . 3 ⊢ (∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ) → ∀y∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ))
3		hbe1 709	. . . . . 6 ⊢ (∃z(A = ⟨y, z⟩ ∧ φ) → ∀z∃z(A = ⟨y, z⟩ ∧ φ))
4	3	hbex 701	. . . . 5 ⊢ (∃y∃z(A = ⟨y, z⟩ ∧ φ) → ∀z∃y∃z(A = ⟨y, z⟩ ∧ φ))
5	4	hbmo 1033	. . . 4 ⊢ (∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ) → ∀z∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ))
6		mosubop.1	. . . . 5 ⊢ ∃*xφ
7		ax-17 925	. . . . . 6 ⊢ (A = ⟨y, z⟩ → ∀x A = ⟨y, z⟩)
8		copsexg 1902	. . . . . 6 ⊢ (A = ⟨y, z⟩ → (φ ↔ ∃y∃z(A = ⟨y, z⟩ ∧ φ)))
9	7, 8	bimod 1030	. . . . 5 ⊢ (A = ⟨y, z⟩ → (∃*xφ ↔ ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)))
10	6, 9	mpbii 168	. . . 4 ⊢ (A = ⟨y, z⟩ → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ))
11	5, 10	19.23ai 746	. . 3 ⊢ (∃z A = ⟨y, z⟩ → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ))
12	2, 11	19.23ai 746	. 2 ⊢ (∃y∃z A = ⟨y, z⟩ → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ))
13		pm3.26 256	. . . . . . 7 ⊢ ((A = ⟨y, z⟩ ∧ φ) → A = ⟨y, z⟩)
14	13	19.22i 723	. . . . . 6 ⊢ (∃z(A = ⟨y, z⟩ ∧ φ) → ∃z A = ⟨y, z⟩)
15	14	19.22i 723	. . . . 5 ⊢ (∃y∃z(A = ⟨y, z⟩ ∧ φ) → ∃y∃z A = ⟨y, z⟩)
16	15	19.23aiv 952	. . . 4 ⊢ (∃x∃y∃z(A = ⟨y, z⟩ ∧ φ) → ∃y∃z A = ⟨y, z⟩)
17	16	con3i 90	. . 3 ⊢ (¬ ∃y∃z A = ⟨y, z⟩ → ¬ ∃x∃y∃z(A = ⟨y, z⟩ ∧ φ))
18		exmo 1042	. . . 4 ⊢ (∃x∃y∃z(A = ⟨y, z⟩ ∧ φ) ∨ ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ))
19	18	ori 200	. . 3 ⊢ (¬ ∃x∃y∃z(A = ⟨y, z⟩ ∧ φ) → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ))
20	17, 19	syl 12	. 2 ⊢ (¬ ∃y∃z A = ⟨y, z⟩ → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ))
21	12, 20	pm2.61i 110	1 ⊢ ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)

Syntax hints:  ¬ wn 1   ∧ wa 196  ∃wex 678  ∃*wmo 1008   = wceq 1091  ⟨cop 1810

This theorem is referenced by:  funoprab 3037  oprabex3 3046  oprabval3 3052

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815

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