Theorem cbvoprab12 3028

Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution.

Hypotheses

Ref	Expression
cbvoprab12.1	⊢ (φ → ∀wφ)
cbvoprab12.2	⊢ (φ → ∀vφ)
cbvoprab12.3	⊢ (ψ → ∀xψ)
cbvoprab12.4	⊢ (ψ → ∀yψ)
cbvoprab12.5	⊢ ((x = w ∧ y = v) → (φ ↔ ψ))

Assertion

Ref	Expression
cbvoprab12	⊢ {⟨⟨x, y⟩, z⟩∣φ} = {⟨⟨w, v⟩, z⟩∣ψ}

Distinct variable group(s):   x,y,z,w,v

Proof of Theorem cbvoprab12

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . . 6 ⊢ (u = ⟨⟨x, y⟩, z⟩ → ∀w u = ⟨⟨x, y⟩, z⟩)
2		cbvoprab12.1	. . . . . 6 ⊢ (φ → ∀wφ)
3	1, 2	hban 704	. . . . 5 ⊢ ((u = ⟨⟨x, y⟩, z⟩ ∧ φ) → ∀w(u = ⟨⟨x, y⟩, z⟩ ∧ φ))
4	3	hbex 701	. . . 4 ⊢ (∃z(u = ⟨⟨x, y⟩, z⟩ ∧ φ) → ∀w∃z(u = ⟨⟨x, y⟩, z⟩ ∧ φ))
5		ax-17 925	. . . . . 6 ⊢ (u = ⟨⟨x, y⟩, z⟩ → ∀v u = ⟨⟨x, y⟩, z⟩)
6		cbvoprab12.2	. . . . . 6 ⊢ (φ → ∀vφ)
7	5, 6	hban 704	. . . . 5 ⊢ ((u = ⟨⟨x, y⟩, z⟩ ∧ φ) → ∀v(u = ⟨⟨x, y⟩, z⟩ ∧ φ))
8	7	hbex 701	. . . 4 ⊢ (∃z(u = ⟨⟨x, y⟩, z⟩ ∧ φ) → ∀v∃z(u = ⟨⟨x, y⟩, z⟩ ∧ φ))
9		ax-17 925	. . . . . 6 ⊢ (u = ⟨⟨w, v⟩, z⟩ → ∀x u = ⟨⟨w, v⟩, z⟩)
10		cbvoprab12.3	. . . . . 6 ⊢ (ψ → ∀xψ)
11	9, 10	hban 704	. . . . 5 ⊢ ((u = ⟨⟨w, v⟩, z⟩ ∧ ψ) → ∀x(u = ⟨⟨w, v⟩, z⟩ ∧ ψ))
12	11	hbex 701	. . . 4 ⊢ (∃z(u = ⟨⟨w, v⟩, z⟩ ∧ ψ) → ∀x∃z(u = ⟨⟨w, v⟩, z⟩ ∧ ψ))
13		ax-17 925	. . . . . 6 ⊢ (u = ⟨⟨w, v⟩, z⟩ → ∀y u = ⟨⟨w, v⟩, z⟩)
14		cbvoprab12.4	. . . . . 6 ⊢ (ψ → ∀yψ)
15	13, 14	hban 704	. . . . 5 ⊢ ((u = ⟨⟨w, v⟩, z⟩ ∧ ψ) → ∀y(u = ⟨⟨w, v⟩, z⟩ ∧ ψ))
16	15	hbex 701	. . . 4 ⊢ (∃z(u = ⟨⟨w, v⟩, z⟩ ∧ ψ) → ∀y∃z(u = ⟨⟨w, v⟩, z⟩ ∧ ψ))
17		opeq12 1878	. . . . . . . 8 ⊢ ((x = w ∧ y = v) → ⟨x, y⟩ = ⟨w, v⟩)
18		opeq1 1876	. . . . . . . 8 ⊢ (⟨x, y⟩ = ⟨w, v⟩ → ⟨⟨x, y⟩, z⟩ = ⟨⟨w, v⟩, z⟩)
19	17, 18	syl 12	. . . . . . 7 ⊢ ((x = w ∧ y = v) → ⟨⟨x, y⟩, z⟩ = ⟨⟨w, v⟩, z⟩)
20	19	cleq2d 1112	. . . . . 6 ⊢ ((x = w ∧ y = v) → (u = ⟨⟨x, y⟩, z⟩ ↔ u = ⟨⟨w, v⟩, z⟩))
21		cbvoprab12.5	. . . . . 6 ⊢ ((x = w ∧ y = v) → (φ ↔ ψ))
22	20, 21	anbi12d 476	. . . . 5 ⊢ ((x = w ∧ y = v) → ((u = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ (u = ⟨⟨w, v⟩, z⟩ ∧ ψ)))
23	22	biexdv 936	. . . 4 ⊢ ((x = w ∧ y = v) → (∃z(u = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ∃z(u = ⟨⟨w, v⟩, z⟩ ∧ ψ)))
24	4, 8, 12, 16, 23	cbvex2 975	. . 3 ⊢ (∃x∃y∃z(u = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ∃w∃v∃z(u = ⟨⟨w, v⟩, z⟩ ∧ ψ))
25	24	biabi 1181	. 2 ⊢ {u∣∃x∃y∃z(u = ⟨⟨x, y⟩, z⟩ ∧ φ)} = {u∣∃w∃v∃z(u = ⟨⟨w, v⟩, z⟩ ∧ ψ)}
26		df-oprab 3004	. 2 ⊢ {⟨⟨x, y⟩, z⟩∣φ} = {u∣∃x∃y∃z(u = ⟨⟨x, y⟩, z⟩ ∧ φ)}
27		df-oprab 3004	. 2 ⊢ {⟨⟨w, v⟩, z⟩∣ψ} = {u∣∃w∃v∃z(u = ⟨⟨w, v⟩, z⟩ ∧ ψ)}
28	25, 26, 27	3eqtr4 1126	1 ⊢ {⟨⟨x, y⟩, z⟩∣φ} = {⟨⟨w, v⟩, z⟩∣ψ}

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  {cab 1090   = wceq 1091  ⟨cop 1810  {copab2 3002

This theorem is referenced by:  cbvoprab12v 3029  oprabval4g 3053

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-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr �1812  df-op 1815  df-oprab 3004

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